Almost sure behavior for the local time of a diffusion in a spectrally negative L\'evy environment

TL;DR

This paper investigates the almost sure asymptotic behavior of the supremum of local time for a diffusion in a spectrally negative Lévy environment, revealing new renormalization results linked to the environment's properties.

Contribution

It provides the first detailed characterization of the extreme asymptotic behavior of local time in this setting, connecting it to the Lévy environment's Laplace exponent.

Findings

Renormalization of large local time values depends on the Lévy environment's Laplace exponent.

In the sub-ballistic regime, the supremum's behavior is linked to the tail of the exponential functional.

The renormalization for small local time values is universal, independent of the environment.

Abstract

We study the almost sure asymptotic behavior of the supremum of the local time for a transient diffusion in a spectrally negative L\'evy environment. More precisely, we provide the proper renormalizations for the extremely large and the extremely small values of the supremum of the local time. In particular, in the sub-ballistic regime, we link the almost sure behavior of the supremum of the local time with the left asymptotic tail of the exponential functional of the environment conditioned to stay positive, which allows to use the properties of such exponential functionals to characterize the sought behavior. It appears from our results that the renormalization of the extremely large values of the supremum of the local time in the sub-ballistic regime is determined by the asymptotic behavior of the Laplace exponent of the L\'evy environment, and is surprisingly greater than the renormalization that was previously known for the recurrent case. For the renormalization of the extremely small values of the supremum of the local time in the sub-ballistic regime, our results show that there is only one possible renormalization, whatever is the choice of the L\'evy environment.

Equations

$$\forall t,\lambda \ge 0,\mathbb{E}\left[e^{\lambda V(t)}\right]=e^{t{\Psi }_V(\lambda )}.$$

$$\text{d}X(t)=\text{d}\beta (t)-\frac{1}{2}{V}^{\prime }(X(t))\text{d}t,$$

$$\frac{1}{2}{e}^{V(x)}\frac{\text{d}}{\text{d}x}\left(e^{-V(x)}\frac{\text{d}}{\text{d}x}\right).$$

$${\mathcal{L}}_X^{\ast }(t)=\underset{x\in \mathbb{R}}{\sup }{\mathcal{L}}_X(t,x).$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t\log (\log (\log (t)))}\ge \frac{1}{32},$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t\log (\log (\log (t)))}\le \frac{e^2}{2}\text{and}\frac{j_0^2}{64}\le \underset{t\to +\infty }{\mathrm{liminf}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t/\log (\log (\log (t)))}\le \frac{e^2{\pi }^2}{4},$$

$$\sum _{n=1}^{+\infty }\frac{1}{na(n)}\{\begin{array}{rl}& <+\infty \\ & =+\infty \end{array}\iff \underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{(ta(t){)}^{1/\kappa }}=\{\begin{array}{rl}& 0\\ & +\infty \end{array}\mathbb{P}\text{-a.s.},$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{liminf}}\frac{{\mathcal{L}}_X^{\ast }(t)}{(t/\log (\log (t)){)}^{1/\kappa }}=4({\kappa }^2(\kappa -1)/8{)}^{1/\kappa }.$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{liminf}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t/\log (t)\log (\log (t))}\le 1/2.$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t}=+\infty ,$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{liminf}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t/\log (\log (t))}\le C(\kappa ),$$

$$\displaystyle\forall\epsilon>0,\$$

$$I(V^{\uparrow }):={\int }_0^{+\infty }{e}^{-V^{\uparrow }(t)}dt\text{and}I({\hat{V}}^{\uparrow }):={\int }_0^{+\infty }{e}^{-{\hat{V}}^{\uparrow }(t)}dt,$$

$$\sigma$$

$$\beta$$

$$\mathbb{P}\left(I(V^{\uparrow })\le x\right)\le e^{-K_0/x}.$$

$$\forall l<\frac{1}{\beta -1},\mathbb{P}\left(I(V^{\uparrow })\le x\right)\le e^{-1/x^l}.$$

$$\text{If}\sigma >1,\forall l>\frac{1}{\sigma -1},\mathbb{P}\left(I(V^{\uparrow })\le x\right)\ge e^{-1/x^l}.$$

$$\exp \left(-\frac{\gamma {\alpha }^{\frac{\alpha }{\alpha -1}}}{(cx{)}^{\frac{1}{\alpha -1}}}\right)\le \mathbb{P}\left(I(V^{\uparrow })\le x\right)\le \exp \left(-\frac{\alpha -1}{\gamma (Cx{)}^{\frac{1}{\alpha -1}}}\right).$$

$$-\log \left(\mathbb{P}\left(I(V^{\uparrow })\le x\right)\right)\underset{x\to 0}{\sim }\frac{\alpha -1}{(Cx{)}^{\frac{1}{\alpha -1}}}.$$

$$\mathbb{P}\left(I(V^{\uparrow })\le x\right)\le \exp \left(-\frac{C}{x^{\frac{1}{\gamma -1}}}\right),$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t(\log (\log (t)){)}^{\gamma -1}}\le C^{1-\gamma }.$$

$$\mathbb{P}\left(I(V^{\uparrow })\le x\right)\ge \exp \left(-\frac{C}{x^{\frac{1}{\gamma -1}}}\right),$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t(\log (\log (t)){)}^{\gamma -1}}\ge C^{1-\gamma }.$$

$$\forall {\beta }^{\prime }>\beta ,\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t(\log (\log (t)){)}^{{\beta }^{\prime }-1}}=0,$$

$$\text{If}\sigma >1,\forall {\sigma }^{\prime }\in ]1,\sigma [,\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t(\log (\log (t)){)}^{{\sigma }^{\prime }-1}}=+\infty .$$

$$\mathbb{P}\text{-a.s.}\frac{c}{{\alpha }^{\alpha }}\le \underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t(\log (\log (t)){)}^{\alpha -1}}\le \frac{C}{(\alpha -1{)}^{\alpha -1}}.$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t(\log (\log (t)){)}^{\alpha -1}}=\frac{C}{(\alpha -1{)}^{\alpha -1}}.$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t(\log (\log (t)))}=\frac{1}{8}.$$

$$\mathbb{P}\text{-a.s.}\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t(\log (\log (t)))}<+\infty .$$

$$\mathbb{P}\text{-a.s.}0<\underset{t\to +\infty }{\mathrm{limsup}}\frac{{\mathcal{L}}_X^{\ast }(t)}{t}<+\infty .$$

Full text

NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China

E-mail address: ghv2@nyu.edu

1. Introduction

We study the almost sure asymptotic behavior of the supremum of the local time for a transient diffusion in a spectrally negative Lévy environment. Let $(V(x),\ x\in\mathbb{R})$ be a spectrally negative Lévy process on $\mathbb{R}$ which is not the opposite of a subordinator (in particular, $V$ can not be a compound Poisson process), drifts to $-\infty$ at $+\infty$, and such that $V(0)=0$. We denote its Laplace exponent by $\Psi_{V}$:

[TABLE]

It is well-known, for such $V$, that $\Psi_{V}$ admits a non trivial zero that we denote here by $\kappa$, $\kappa:=\inf\{\lambda>0,\ \Psi_{V}(\lambda)=0\}>0$.

We are here interested in a diffusion in this potential $V$. Such a diffusion $(X(t),\ t\geq 0)$ is defined informally by $X(0)=0$ and

[TABLE]

where $\beta$ is a Brownian motion independent from $V$. Rigorously, $X$ is defined by its conditional generator given $V$,

[TABLE]

Diffusions in random potentials have been introduced, in the case of a brownian potential, by Schumacher [35] and Brox [9]. They are generally considered as the continuous time analogue of Random Walks in Random Environments (RWRE), which have many applications in physics and biology (see for example Le Doussal and al. [19]). We refer to Révész [34] and Zeitouni [47] for general properties of RWRE and to Shi [37] for the embedding of a RWRE into a diffusion in random potential. In the case of a drifted brownian potential, the diffusion $X$ has first been studied by Kawazu, Tanaka [29] who exhibit three different regimes. These results have then been refined by Kawazu, Tanaka [30], Tanaka [44], and Hu, Shi and Yor [28] who, in the ballistic case, exhibit several possible behaviors and prove theorems of type central limit. Large deviations results are obtained in Taleb [41], Talet [42], and moderate deviations are given by Hu, Shi [27] in the recurrent case, and by Faraud [23] in the transient case. A result on localization and an aging phenomenon are proved in Andreoletti, Devulder [4] for the case of a drifted Brownian potential in the sub-ballistic regime. The diffusion has been studied in the case of a Lévy potential by Carmona [10], Cheliotis [11] and Singh [40], [38], [39]. Let us also mention that diffusions in random potentials have been studied in dimension higher than $1$ by Tanaka [43] and Mathieu [32], [33].

In our case of a spectrally negative Lévy potential $V$ drifting to $-\infty$, the fact that $V$ drifts to $-\infty$ puts us in the case where the diffusion $X$ is a.s. transient to the right. The asymptotic behavior of this diffusion has been studied by Singh [40], he distinguishes three main possible behaviors depending on $0<\kappa<1$, $\kappa=1$ or $\kappa>1$ (the case $\kappa>1$ being also divided into three sub-cases). We denote by $(\mathcal{L}_{X}(t,x),t\geq 0,x\in\mathbb{R})$ the version of the local time that is continuous in time and càd-làg in space, and we define the supremum of the local time until instant $t$ as

[TABLE]

Here, we study the almost sure asymptotic behavior of $\mathcal{L}_{X}^{*}(t)$. The analogous problems for RWRE have attracted much attention, and have been studied, for example, in Révész [34], Shi [36], Gantert and al. [24], [25], Hu and al. [26], Dembo and al. [13] and Andreoletti [2], [1]. The local time of processes in random environment plays an important role in the problem of estimating the potential (Comets and al. [12], Andreoletti [3]), in the study of persistence (see Devulder [15]) and is useful for the study of related models like random walks in stratified random environment (see Kochler [31]) or processes in random scenery (see Zindy [48]). When the environment is a Brownian motion with no drift (in this case the diffusion is recurrent), Shi [36] has studied the behavior of $\mathcal{L}_{X}^{*}(t)$ and shown that

[TABLE]

where $\mathbb{P}$ is the so-called annealed probability measure whose definition is recalled in Subsection 1.3. In the same case, Andreoletti and Diel [6] have proved more recently the convergence in distribution of $\mathcal{L}_{X}^{*}(t)/t$ under $\mathbb{P}$. Diel [16] has then continued the study by giving a finite upper bound for the $\limsup$ in (1.1) and doing the same study for the $\liminf$:

[TABLE]

where $j_{0}$ is the smallest positive root of the Bessel function $J_{0}$.

For a drifted Brownian environment (in this case the diffusion is transient), the convergence in distribution of $\mathcal{L}_{X}^{*}(t)/t$ under $\mathbb{P}$ has been established by Andreoletti, Devulder and Véchambre [5] in the sub-ballistic case $0<\kappa<1$ and by Devulder [14] in the case $\kappa\geq 1$. Still in the case of a drifted Brownian environment, the almost sure behavior of $\mathcal{L}_{X}^{*}(t)$ has been studied via annealed methods in [14], where this behavior is totally characterized in the ballistic case $\kappa>1$. In this case, for any positive non-decreasing function $a$ we have

[TABLE]

and

[TABLE]

When $\kappa=1$ he obtains

[TABLE]

In the sub-ballistic case $0<\kappa<1$, his method fails and only provides partial results for the almost sure behavior of the local time. More precisely he proves that the renormalization for the $\limsup$ is greater than $t$:

[TABLE]

and that the renormalization for the $\liminf$ is at most $t/\log(\log(t))$ and greater than

$t/(\log(t))^{1/\kappa}(\log(\log(t)))^{(2/\kappa)+\epsilon}$ for any $\epsilon>0$:

[TABLE]

where $C(\kappa)$ is a positive non explicit constant. The main motivation of the present paper is to provide new methods that allow to determine precisely the almost sure asymptotic behavior of the local time in the case $0<\kappa<1$ for general spectrally negative Lévy environments.

For the (discrete) transient RWRE, the almost sure behavior of the supremum of the local time has been studied by Gantert and Shi [25]. They obtain the behavior of the $\limsup$ in the two sub-cases $0<\kappa\leq 1$ and $\kappa>1$ but no result has been established so far for the $\liminf$.

For the recurrent diffusion in Lévy potentials, the convergence in distribution of $\mathcal{L}_{X}^{*}(t)/t$ under $\mathbb{P}$ has been established by Diel and Voisin [17] in the case of a stable Lévy environment. For the transient diffusion in a general spectrally negative Lévy potential $V$, the convergence in distribution of $\mathcal{L}_{X}^{*}(t)/t$ under $\mathbb{P}$ has been established by the author in [46] using two different methods: when $0<\kappa<1$ a path decomposition of the environment that provides an interesting renewal structure to study the diffusion is used, this method is inspired from Andreoletti and al. [5], [4] which are themselves inspired from the work of Enriquez and al. [21] in the discrete case. When $\kappa>1$ an equality in law between the local time and a generalized Ornstein-Uhlenbeck process that was introduced in [40] is used. The main contribution of the present paper is in the case $0<\kappa<1$. We make an optimized use of the renewal structure of the diffusion, working sometimes in a quenched setting, in order to relate the almost sure asymptotic behavior of $\mathcal{L}_{X}^{*}(t)$ with the asymptotic behavior of some functionals of the renewal structure (this requires to overcome the difficulties caused by the "last valley" as explained in Section 1.2 below), and to establish precisely the asymptotic behavior of these functionals. We characterize the asymptotic almost sure behavior of $\mathcal{L}_{X}^{*}(t)$ when $0<\kappa<1$ and $\kappa>1$. In particular, the restriction of our results to the case of a drifted Brownian potential with $0<\kappa<1$ considerably improves the results (1.5), (1.6) and (1.7) of [14] by giving the exact renormalizations for the $\limsup$ and the $\liminf$, and even the exact value of the $\limsup$. We also get an explicit upper bound for the value of the $\liminf$.

1.1. Main results

We start with the case $0<\kappa<1$. In that case, the limit distribution of $\mathcal{L}_{X}^{*}(t)/t$ under $\mathbb{P}$ is given in Theorem 1.3 of [46] and it depends on exponential functionals of $V$ and its dual conditioned to stay positive. They are defined as follow:

[TABLE]

where $\hat{V}$ denotes the dual of $V$ (it is equal in law to $-V$), and where $V^{\uparrow}$ and $\hat{V}^{\uparrow}$ denote respectively $V$ and $\hat{V}$ conditioned to stay positive. In Subsection 4.1 it is recalled how $V^{\uparrow}$ and $\hat{V}^{\uparrow}$ are defined rigorously. These functionals are studied by the author in [45] where it is proved in Theorems 1.1 and 1.13 that they are indeed finite and well-defined. Let $G_{1}$ and $G_{2}$ be two independent random variables with $G_{1}\overset{\mathcal{L}}{=}I(V^{\uparrow})$ and $G_{2}\overset{\mathcal{L}}{=}I(\hat{V}^{\uparrow})$. We define $\mathcal{R}:=G_{1}+G_{2}$.

To study the $\limsup$ of $\mathcal{L}_{X}^{*}(t)$, we link the almost sur asymptotic behavior of $\mathcal{L}^{*}_{X}(t)$ with the left tail of $I(V^{\uparrow})$ (or of $\mathcal{R}$ depending on the case). Before stating our results, let us recall what is known about the left tail of $I(V^{\uparrow})$.

In [45], the left tail of $I(V^{\uparrow})$ is linked to the asymptotic behavior of $\Psi_{V}$. This asymptotic behavior is usually quantified thanks to two real numbers $\sigma$ and $\beta$:

[TABLE]

If $\Psi_{V}$ has $\alpha$-regular variation for $\alpha\in[1,2]$ (for example if $V$ is a drifted $\alpha$-stable Lévy process with no positive jumps), we have $\sigma=\beta=\alpha$. Note that when $Q$, the Brownian component of $V$, is positive, then $\Psi_{V}$ has $2$-regular variation, and when $Q=0$, $1\leq\sigma\leq\beta\leq 2$. The asymptotic behavior of $\mathbb{P}(I(V^{\uparrow})\leq x)$ as $x$ goes to [math] is given by the following results from [45]:

Theorem 1.1**.**

[Véchambre, [45]]

There is a positive constant $K_{0}$ (depending on $V$) such that for $x$ small enough

[TABLE]

More precisely we have

[TABLE]

[TABLE]

If there are two positive constants $c<C$ and $\alpha\in]1,2]$ such that $c\lambda^{\alpha}\leq\Psi_{V}(\lambda)\leq C\lambda^{\alpha}$ for $\lambda$ large enough, then for any $\gamma>1$ we have, when $x$ is small enough,

[TABLE]

If there is a positive constant $C$ and $\alpha\in]1,2]$ such that $\Psi_{V}(\lambda)\sim_{\lambda\rightarrow+\infty}C\lambda^{\alpha}$, then

[TABLE]

(1.8) is Remark 1.7 of [45], (1.9) and (1.10) are a reformulation of Theorem 1.4 of [45], (1.11) comes from Theorem 1.2 of [45], and (1.12) is Corollary 1.6 of [45].

We can now state our results for the $\limsup$:

Theorem 1.2**.**

• •

Assume that $0<\kappa<1$, $V$ has unbounded variation, $V(1)\in L^{p}$ for some $p>1$ and $V$ possesses negative jumps.

If there is $\gamma>1$ and $C>0$ such that for $x$ small enough

[TABLE]

then we have

[TABLE]

If there is $\gamma>1$ and $C>0$ such that for $x$ small enough

[TABLE]

then we have

[TABLE]

• •

Assume now that $V(t)=W_{\kappa}(t):=W(t)-\frac{\kappa}{2}t$ with $0<\kappa<1$, (i.e. $V$ is the $\kappa$-drifted Brownian motion), then the above implications ((1.13) $\Rightarrow$ (1.14) and (1.15) $\Rightarrow$ (1.16)) are still true with $I(V^{\uparrow})$ replaced by $\mathcal{R}$.

In the above theorem we had to distinguish the case where $V$ possesses negative jumps and the case where $V$ is a drifted Brownian motion. First, note that this is a true alternative: since $V$ is spectrally negative, the case where $V$ does not possess negative jumps is the case where $V$ does not possess jumps at all and is therefore a drifted Brownian motion. In this case we assumed for convenience that the gaussian component of $V$ is normalized to $1$. The difference between the two cases in the above theorem comes from the absence or presence of symmetry for the environment. When $V$ possesses negative jumps, $\hat{V}^{\uparrow}$ possesses positives jumps that may repulse it very fast from [math], this is why the left tail of $I(V^{\uparrow})$ is thiner than the left tail of $I(\hat{V}^{\uparrow})$, as we can see from the results of [45] (comparing Remark 1.7 of [45] with the combination of Theorems 1.15 and 1.16 of [45]). As a consequence, only the left tail of $I(V^{\uparrow})$ is relevant in the left tail of $\mathcal{R}$. When $V$ is a drifted Brownian motion, a symmetry appears: $\hat{V}^{\uparrow}$ and $V^{\uparrow}$ are equal in law, $\mathcal{R}$ is then the sum of two independent random variables having the same law as $I(V^{\uparrow})$ and none of them can be neglected.

Remark 1.3**.**

It has to be noted that the $\limsup$ above is $\mathbb{P}$-almost surely equal to a constant belonging to $[0,+\infty]$ and that the inequalities (1.14) and (1.16) are inequalities relative to this constant that the $\limsup$ equals $\mathbb{P}$-almost surely. The same will be true in all the results below: all the $\limsup$ and $\liminf$ considered are $\mathbb{P}$-almost surely equal to constants. This fact is justified in Subsection 4.4.

Putting together Theorem 1.2 and what is known for the left tail of $I(V^{\uparrow})$ (Theorem 1.1), we can state precise results for the $\limsup$:

Theorem 1.4**.**

If $0<\kappa<1$, $V$ has unbounded variation and $V(1)\in L^{p}$ for some $p>1$, then we have

[TABLE]

and

[TABLE]

If we make further hypothesis on the regularity of the variation of $\Psi_{V}$ we can give the exact order of $\mathcal{L}^{*}_{X}(t)$:

Theorem 1.5**.**

• •

Assume that $0<\kappa<1$, $V$ has unbounded variation, $V(1)\in L^{p}$ for some $p>1$ and $V$ possesses negative jumps.

If there are two positive constants $c<C$ and $\alpha\in]1,2]$ such that $c\lambda^{\alpha}\leq\Psi_{V}(\lambda)\leq C\lambda^{\alpha}$ for $\lambda$ large enough, then we have

[TABLE]

If, more precisely, there is a positive constant $C$ and $\alpha\in]1,2]$ such that $\Psi_{V}(\lambda)\sim C\lambda^{\alpha}$ for large $\lambda$, then we have

[TABLE]

• •

Assume now that $V=W_{\kappa}$, the $\kappa$-drifted Brownian motion, with $0<\kappa<1$, then we have

[TABLE]

Remark 1.6**.**

According to the combination of Theorem 1.2 and (1.8) we see that, if $0<\kappa<1$, $V$ has unbounded variation and $V(1)\in L^{p}$ for some $p>1$, then we always have

[TABLE]

In other words, $t(\log(\log(t)))$ is the maximal possible renormalization for the $\limsup$.

Remark 1.7**.**

As it was noticed by Shi [36] and Diel [16] for the recurrent case, we also notice a difference between the renormalization of the local time for the discrete transient RWRE with zero speed (given by Gantert and Shi [25]) and the renormalization of the local time that we give for the transient diffusion with zero speed. This difference can be explained as in the recurrent case: the valleys can potentially be much steeper in the continuous case with a potential having unbounded variation than in the discrete case, so the local maxima of the local time can potentially be higher in the first case.

We see in the above two theorems that the renormalization of $\mathcal{L}^{*}_{X}(t)$ for the $\limsup$ depends directly on the asymptotic behavior of $\Psi_{V}$. In particular, Theorem 1.5 says that for a drifted $\alpha$-stable environment (with $\alpha>1$, no positive jumps, and the drift is small negative), the renormalization of $\mathcal{L}^{*}_{X}(t)$ is $t(\log(\log(t)))^{\alpha-1}$. We thus see that with general spectrally negative Lévy environments, we have many possible behaviors for $\mathcal{L}^{*}_{X}(t)$, whereas with drifted Brownian environments the only possible renormalization for the $\limsup$ is $t(\log(\log(t)))$. This variety of behaviors is an important motivation to make the study for general spectrally negative Lévy environments.

It is known that the behavior of the diffusion $X$ is related to the right tail of the exponential functional $\int_{0}^{+\infty}e^{V(s)}ds$, and such results also exist in the discrete case (see for example [22], [20]). A novelty in the present paper is to show, in Theorem 1.2, that the asymptotic almost sure behavior of $\mathcal{L}^{*}_{X}(t)$ is crucially related to the left tail of another exponential functional, $I(V^{\uparrow})$, which allows to obtain optimal results for the $\limsup$, in Theorem 1.5, when $\Psi_{V}$ is nice. Even though the distribution of $I(V^{\uparrow})$ appeared in a complicated way in the limit distribution of $\mathcal{L}^{*}_{X}(t)/t$ in [46], it is a surprising fact that the distribution of $I(V^{\uparrow})$ (more precisely its left tail) also totally determines the asymptotic behavior of the extremely large values of the local time.

Even if, for technical reasons, the above theorems do not apply when the environment $V$ has bounded variation, we can conjecture that the behavior of $\mathcal{L}^{*}_{X}(t)$ remains linked in the same way to the left tail of $I(V^{\uparrow})$ which is given by Remark 1.8 of [45]. This would imply

Conjecture 1.8**.**

If $0<\kappa<1$ and $V$ has bounded variation, we have

[TABLE]

If this conjecture is true, we would have, when the environment has bounded variation, the same renormalization as in the discrete transient case given by Theorem 1.1 of [25]. This would not be surprising since the discrete case gives rise to potentials of bounded variation. Moreover, if $V$ has bounded variation then it is known to be the difference of a deterministic positive drift and a subordinator. The valleys can then not be steeper than the deterministic drift so, according to Remark 1.7, the expected renormalization of $\mathcal{L}^{*}_{X}(t)$ has to be the same as in the discrete case.

For the $\liminf$, there is only one possible renormalization. Our result is as follows:

Theorem 1.9**.**

If $0<\kappa<1$, $V$ has unbounded variation and $V(1)\in L^{p}$ for some $p>1$, then we have

[TABLE]

Note that the expectations $\mathbb{E}[I(V^{\uparrow})]$ and $\mathbb{E}[I(\hat{V}^{\uparrow})]$ are finite and well defined since $I(V^{\uparrow})$ and $I(\hat{V}^{\uparrow})$ both admit some finite exponential moments according to Theorems 1.1 and 1.13 of [45].

Example: We consider $W_{\kappa}$, the $\kappa$-drifted Brownian motion ($W_{\kappa}(t):=W(t)-\frac{\kappa}{2}t$), then, the expression of the Laplace transform of $I(W_{\kappa}^{\uparrow})$ is given by equation $(1.12)$ of [45]. This expression allows to compute the moments of $I(W_{\kappa}^{\uparrow})$ and gives in particular $\mathbb{E}[I(W_{\kappa}^{\uparrow})]=2/(1+\kappa)$. Moreover $W_{\kappa}^{\uparrow}$ and $\hat{W}_{\kappa}^{\uparrow}$ have the same law so $\mathbb{E}[I(W_{\kappa}^{\uparrow})]+\mathbb{E}[I(\hat{W}_{\kappa}^{\uparrow})]=4/(1+\kappa)$. If we choose, as an environment, $V=W_{\kappa}$ (for $0<\kappa<1$), then the above upper bound for the $\liminf$ becomes $(1-\kappa^{2})/4\kappa$. Putting this in relation with the results of [14], we see that the application of Theorem 1.9 to the special case of a drifted Brownian environment improves (1.7) and completes (1.6) by proving that this renormalization is exact and by providing an explicit upper bound.

The fact that we have many possible renormalizations for the $\limsup$, depending on the environment $V$, while only one possible for the $\liminf$, whatever is the environment $V$, might seem surprising, here is an heuristic explanation: In each valley, the time spent equals approximately the contribution to the local time multiplied by an exponential functional of the bottom of the valley (which is close to $\mathcal{R}$). The $\limsup$ involves large values of the local time at a fixed time, it is reached when the contribution to the local time in some valley is large while the time spent in the same valley is around a fixed value, this happens when the exponential functional of the bottom of the valley has a small value. The link between the $\limsup$ and the small values of an exponential functional is stated formally in Theorem 1.2. The $\liminf$ involves small values of the local time at a fixed time, it is reached when the contributions to the local time of the valleys are small while the sum of their contributions to the time spent is around a fixed value, this happens when the exponential functionals of the bottoms of some valleys are large. We see that the difference between $\limsup$ and $\liminf$ comes from the difference between the left and right tails of $\mathcal{R}$. The left tail is mainly the left tail of $I(V^{\uparrow})$ which depends on the asymptotic of $\Psi_{V}$, according to Theorem 1.1, and $\Psi_{V}$ has many possible behaviors depending on the choice of $V$. On the other hand, the right tail is always exponential according to Theorems 1.1 and 1.13 of [45]. This explains the difference of behaviors between the $\limsup$ and the $\liminf$.

We now treat the case $\kappa>1$. In this case the limit distribution of $\mathcal{L}_{X}^{*}(t)/t^{1/\kappa}$ under $\mathbb{P}$ is given in Theorem 1.1 of [46]. The methods and the results are different from the case $0<\kappa<1$. For the almost behavior we have

Theorem 1.10**.**

Let $f$ be a positive non-increasing function. When $\kappa>1$, we have

[TABLE]

The above result is the analogue of Theorem 1.2 of [25] for the continuous case. In the special case where $V=W_{\kappa}$ (for $\kappa>1$), our result coincides with (1.3) (proved by Devulder in [14]). Indeed, since $f$ is decreasing we have the easy equivalences

[TABLE]

The first equivalence shows that our result agrees with (1.3) and the second equivalence allows to reformulate the integrability condition in a form that is convenient to prove Theorem 1.10.

Remark 1.11**.**

Comparing Theorems 1.4, 1.5 and 1.10 we see that the renormalization for the $\limsup$ is larger in the transient sub-ballistic regime than in the transient ballistic regime, which is in accordance with intuition. However, it is surprising to see that the renormalization in the transient sub-ballistic regime is also greater than the renormalization in the recurrent case, given by (1.1)-(1.2) which is Theorem 1.1 of [16]. Here is the heuristic explanation: In the recurrent case the diffusion is trapped in the bottom of a large valley while in the transient sub-ballistic regime the diffusion gets successively trapped in the bottom of many valleys, their bottoms being much more narrow. This explains that the large values of the local time have the tendency to be higher in the second case.

For the $\liminf$, we provide an explicit value. Let us define the constants $K$ and $m$ similarly as in [40]:

[TABLE]

We see that $m$ is properly defined when $\kappa>1$. We have:

Theorem 1.12**.**

When $\kappa>1$, we have

[TABLE]

Example: If we choose $V=W_{\kappa}$ (for $\kappa>1$), then $K=2^{\kappa-1}/\Gamma(\kappa)$ (see Example 1.1 in [40]) and $m=4/(\kappa-1)$. The above limit is then $4(\kappa^{2}(\kappa-1)/8)^{1/\kappa}$. This coincides with (1.4) (proved by Devulder in [14]).

For the $\liminf$ the behaviors of the different cases are in accordance with intuition: comparing Theorem 1.1 of [16] with Theorems 1.9 and 1.12 we see that the renormalization for the $\liminf$ in the transient ballistic regime is smaller than the renormalization in the transient sub-ballistic regime which is in turn smaller than the renormalization in the recurrent case.

The assumption that $V$ is spectrally negative is important in this work for several reasons: it allows the Laplace exponent $\Psi_{V}$ and its non-trivial zero $\kappa$ to be defined and the right tail of the exponential functional $\int_{0}^{+\infty}e^{V(u)}du$ to be explicit, $V^{\uparrow}$ is well defined, has infinite life-time, and is related to the law of ascending valleys in a simple way, the left tail of $I(V^{\uparrow})$, which is linked to the asymptotic behavior of $\mathcal{L}^{*}_{X}(t)$ when $0<\kappa<1$, is explicit and particularly interesting in the spectrally negative case, finally some estimates are easier to prove in the spectrally negative case.

1.2. Sketch of proofs and organisation of the paper

The methods used in the present paper for the case $0<\kappa<1$ bear significant improvements compared with the methods used in [5] and [46]. Since the present paper studies the almost sure behavior, estimates need to be more precise which turns out to be a serious obstacle when one needs to deal with the last valley, that is, the valley where the diffusion is stuck at instant $t$. In general, what happens inside this last valley is not easy to quantify since for this valley we have to look at the contribution of the valley until time $t$ instead of the total contribution of the valley, this truncated peak of local time is difficult to estimate and is not independent from the contributions of the previous valleys. Moreover, a difficulty comes from the fact that this valley and what happens inside it is different from other valleys precisely because it is "a valley that is conditioned to trap the diffusion for a substantial amount of time". This last difficulty can be overcome by discretizing on the possible indices for the last valley but this requires to prove that precise estimates hold for a large number of valleys simultaneously. In order to get fine estimates we here extend the methods from [5] and [46] to a "more quenched" setting: in Section 2 we define carefully a set $\mathcal{G}_{t}$ of good environments that has high probability and we establish in Fact 2.7 that for all good environments, an inequality satisfied by the truncated peak of local time in the $k^{th}$ valley corresponds with high quenched probability to an inequality for the random variable $R_{k}^{t}$ that represent the exponential functional of the bottom of the valley (more precisely, Fact 2.7 says that for good environments the quenched probability that the $k^{th}$ peak of local time satisfies some inequality while $R_{k}^{t}$ does not satisfy the corresponding inequality is small). Then, having such estimates available in the quenched setting allows to make a better use of the Markov property, which allows to get more independence and to optimize the use of the renewal structure for the diffusion. A surprising fact that is that, even though the contribution of the last valley appears in the limit distribution of $\mathcal{L}^{*}_{X}(t)/t$, the method just described allows us to prove Lemma 2.14 which shows that this contribution can be neglected when we deal with the extremely large values of the local time, and this is the reason why we can get optimal results for the $\limsup$. For the extremely small values, the method allows to control the influence of the last valley in the proof of Proposition 2.17, but the last valley cannot be totally neglected and stays in the way, this is one of the reasons why the results for the $\liminf$ are a little less precise. Another reason is that the behavior of the extremely small values of the local time is related to the right distribution tail of $\mathcal{R}$ but the later is not precisely known.

Once the effect of the last valley has been removed from the study, as described above, we can prove Propositions 2.11 and 2.17, which transforms the problem into the study of the extreme behaviors of functionals (of the renewal structure) that represent the highest peak of local time among all valleys before the last valley (the peak of local time in the last valley is not taken into account for $Y_{1}^{\natural,t}(Y_{2}^{-1,t}(1)-)$, the functional involved in Proposition 2.11, but it is for $Y_{1}^{\natural,t}(Y_{2}^{-1,t}(1))$, the functional involved in Proposition 2.17). The extreme behaviors of these functionals are obtained in Propositions 2.15 and 2.20. It can easily be seen from the results of [46] that these functionals converge in distribution to functionals of a $\kappa$-stable bivariate subordinator. However, the extreme behaviors of these functionals had not been studied before and cannot be identified trivially. Our first step to prove Propositions 2.15 and 2.20 is to reverse the problem: instead of considering the highest peak of local time until the sum of the contributions to the time spent by the diffusion in the successive valleys becomes greater than $t$, we consider the sum of the contributions to the time until the first peak of local time higher than some extremely large or small value. For example, for Proposition 2.15, we will need to study the event $\{Y_{1}^{\natural,t}(Y_{2}^{-1,t}(1)-)\geq y_{t}\}$ (for $y_{t}$ a large positive number depending on $t$) which represents the fact that, before the sum of the contributions to the time spent by the diffusion in the successive valleys becomes greater than $t$, there is a peak of local time that is greater than $y_{t}$. This event is equal to the event where, at the first peak of local time higher than $y_{t}$, the sum of the contributions to the time spent by the diffusion in the successive valleys is less than $t$. We thus need to deal with two quantities: 1) the sum of the contributions to the time spent by the diffusion in the successive valleys before $-$ and not including $-$ the valley of the first peak of local time higher than $y_{t}$, 2) the contribution to the time spent in the valley of the first peak of local time higher than $y_{t}$. For Proposition 2.20, only a quantity similar to 1) appears. In any case, the index of the first peak of local time higher than $y_{t}$ (or any other fixed value) follows a geometric distribution and the time contributions for 1) and 2) are obtained by taking "normal valleys" conditioned on the fact that their peak of local time is, respectively, smaller and higher than $y_{t}$, which allows to study their extreme behaviors either directly or via the asymptotic behavior of their Laplace transform. Results on the Laplace transform of these particular time contributions are established in Lemmas 2.18 and 2.19. In order to proves these lemmas (and, in the case of the $\limsup$, to understand directly the extreme behavior of the quantity 2)), we fully exploit the precise knowledge of the extreme behavior of the peak of local time in a normal valley, which was established in [46] as an intermediary result and is recalled in Fact 2.8 below.

The rest of the paper is organized as follows.

In Section 2 we study the case $0<\kappa<1$. We first recall the decomposition of the environment into valleys from [46] and the behavior of the diffusion with respect to these valleys. In particular, we recall how the renewal structure of the diffusion allows to approximate the supremum of the local time and the time spent by the diffusion in the bottom of the valleys by an iid sequence of $\mathbb{R}_{+}^{2}$-valued random variables. We also define the set $\mathcal{G}_{t}$ of good environments and state Fact 2.7 mentioned above.

For the $\limsup$, we study the asymptotic of the probability $\mathbb{P}(\mathcal{L}^{*}_{X}(t)/t\geq x_{t})$, where $x_{t}$ is a suitably chosen quantity that goes to infinity with $t$. More precisely, we prove Lemma 2.14 which allows to neglect the effect of the last valley, and we can then prove Proposition 2.11 which relates very precisely the asymptotic of the probability $\mathbb{P}(\mathcal{L}^{*}_{X}(t)/t\geq x_{t})$ with the one of $\mathbb{P}(Y_{1}^{\natural,t}(Y_{2}^{-1,t}(1)-)\geq x_{t})$. In Proposition 2.15 we link the asymptotic behavior of $\mathbb{P}(Y_{1}^{\natural,t}(Y_{2}^{-1,t}(1)-)\geq x_{t})$ with the left tail of $I(V^{\uparrow})$ (or $\mathcal{R}$ in the case of a drifted Brownian potential). The synthesis of Propositions 2.11 and 2.15 allows to compare the behavior of $\mathbb{P}(\mathcal{L}^{*}_{X}(t)/t\geq x_{t})$ with the left tail of $I(V^{\uparrow})$ (or $\mathcal{R}$ in the case of a drifted Brownian potential) in Proposition 2.16. This proposition entails Theorem 1.2 by the mean of the Borel-Cantelli Lemma and of the technical Lemma 2.10, which decomposes the trajectory of the diffusion into large independent parts in order to get the required independence to apply the Borel-Cantelli Lemma. The combination of Theorem 1.2 with what is known (and recalled in Theorem 1.1) for the left tail of $I(V^{\uparrow})$ easily yields Theorems 1.4 and 1.5, which solves the problem for the $\limsup$.

For the $\liminf$, we study the quantity $\mathbb{P}(\mathcal{L}^{*}_{X}(t)\leq t/x_{t})$. In Proposition 2.17 it is compared with $\mathbb{P}(Y_{1}^{\natural,t}(Y_{2}^{-1,t}(1))\leq 1/x_{t})$. Taking care of the last valley is done directly in the proof of Proposition 2.17, with the method described in the beginning of this subsection. In Lemmas 2.18 and 2.19 we study the Laplace transform of a random variable involved in this functional, as explained above. This allows to give a lower and an upper bound for $\mathbb{P}(Y_{1}^{\natural,t}(Y_{2}^{-1,t}(1))\leq 1/x_{t})$ in Proposition 2.20. The synthesis of Propositions 2.17 and 2.20 gives the asymptotic of $\mathbb{P}(\mathcal{L}^{*}_{X}(t)\leq t/x_{t})$ in Proposition 2.21. This Proposition entails Theorem 1.9 by the mean of the Borel-Cantelli Lemma and Lemma 2.10. This solves the problem for the $\liminf$.

In Section 3 we study the case $\kappa>1$. In this case, the local time at $t$ can be approximated by the local time at a hitting time and the latter has the same law as the generalized Ornstein-Uhlenbeck process introduced in [40]. Using what is known for the excursion measure of this process we prove Theorems 1.10 and 1.12.

In Section 4 we justify some facts about $V$, $V^{\uparrow}$ and the diffusion in $V$ that are used along the paper.

1.3. Facts and notations

For $Y$ a process and $S$ a borelian set, we denote

[TABLE]

We shall only write $\tau(Y,x)$ instead of $\tau(Y,\{x\})$ and $\tau(Y,x+)$ instead of $\tau(Y,[x,+\infty[)$. Since $V$ has no positive jumps we see that each positive level is reached continuously (or not reached at all): $\forall x>0,\ \tau(V,x+)=\tau(V,x)$ (which is possibly infinite). Moreover, the law of the supremum of $V$ is known, it is an exponential distribution with parameter $\kappa$ (see Corollary VII.2 in Bertoin [7]).

If $Y$ is Markovian and $x\in\mathbb{R}$ we denote $Y_{x}$ for the process $Y$ starting from $x$. For $Y_{0}$ we shall only write $Y$. When it exists we denote by $(\mathcal{L}_{Y}(t,x),t\geq 0,x\in\mathbb{R})$ the version of the local time of $Y$ that is continuous in time and càd-làg in space and by $(\sigma_{Y}(t,x),t\geq 0,x\in\mathbb{R})$ the inverse of this local time: $\sigma_{Y}(t,x):=\inf\{s\geq 0,\ \mathcal{L}_{Y}(s,x)>t\}$.

Let $B$ be a Brownian motion starting at [math] and independent from $V$. A diffusion in potential $V$ can be defined via the formula

[TABLE]

where

[TABLE]

$A_{V}$ is the scale function of the diffusion $X$ (at fixed environment $V$). Note that, by the time-reversal property for Lévy processes, the process $(-V(-(x-)),\ x\geq 0)$ has the same law as $(V(x),\ x\geq 0)$ so $V$ converges almost surely to $+\infty$ at $-\infty$. In particular we have $A_{V}(-\infty)=-\infty$ while $A_{V}(+\infty)<+\infty$. This implies that $X$ is almost surely transient toward $+\infty$.

It is known that the local time of $X$ at $x$ until instant $t$ has the following expression:

[TABLE]

Recall the notation $\mathcal{L}^{*}_{X}(t)$ for the overall supremum of the local time until time $t$. We use the notation $\mathcal{L}_{X}^{*,+}$ for the supremum of the local time on the positive half-line:

[TABLE]

For the hitting times of $r\in\mathbb{R}$ by the diffusion $X$ we shall use the frequent notation $H(r)$ (instead of $\tau(X,r)$).

$D(\mathbb{R},\mathbb{R})$ is the space of càd-làg functions from $\mathbb{R}$ to $\mathbb{R}$. Let $P$ be the probability measure on $D(\mathbb{R},\mathbb{R})$ inducing the law of $V$. For $v\in D(\mathbb{R},\mathbb{R})$, the quenched probability measure $P^{v}$ is the probability measure (associated with the diffusion $X$) conditionally on $\{V=v\}$. $\mathbb{P}$ represents the annealed probability measure, it is defined as $\mathbb{P}(.):=\int_{D(\mathbb{R},\mathbb{R})}P^{v}(.)P(dv)$. $X$ is a Markovian process under $P^{v}$ but not under $\mathbb{P}$. Note that all the almost sure convergences stated in this Introduction are $\mathbb{P}$-almost sure convergences. For objects not related to the diffusion $X$ we also use the natural notation $\mathbb{P}$ for a probability.

If $Z$ is a random variable, its law is denoted by $\mathcal{L}(Z)$ and if $A$ is an event of positive probability, $\mathcal{L}(Z|A)$ denotes the law of $Z$ conditionally to the event $A$.

For $Z$ an increasing càd-làg process and $s\geq 0$, we put respectively $Z(s-)$, $Z^{\natural}(s)$ and $Z^{-1}(s)$ for respectively the left-limit of $Z$ at $s$, the largest jump of $Z$ before $s$ and the generalized inverse of $Z$ at $s$:

[TABLE]

For two quantities $a$ and $b$ depending on a parameter, $a\approx b$ means that $\log(a)\sim\log(b)$ when the parameter converges (generally to [math] or infinity).

2. Almost sure behavior when $0<\kappa<1$

In all this section we assume that the hypotheses of Theorems 1.4 and 1.9 are satisfied: $0<\kappa<1$, $V$ has unbounded variation and there exists $p>1$ such that $V(1)\in L^{p}$. The hypothesis of unbounded variation is necessary to approximate the law of the left part of a valley by the law of $\hat{V}^{\uparrow}$ and the hypothesis about moments for $V(1)$ allows to neglect the local time outside the bottoms of the valleys. For these reasons, many results of [46] (that are recalled in the next subsection) have been proved under these hypotheses.

2.1. Traps for the diffusion

We now recall some definitions about valleys and describe how the diffusion gets trapped into successive valleys. The facts and lemmas stated in this subsection are more or less classical and there are all proved or justified in Subsection 4.3 (except Fact 2.3 which is readily Lemma 3.5 of [46]).

We first recall the notion of $h$-extrema. For $h>0$, we say that $x\in\mathbb{R}$ is an $h$-minimum for $V$ if there exist $u<x<v$ such that $V(y)\wedge V(y-)\geq V(x)\wedge V(x-)$ for all $y\in[u,v]$, $V(u)\geq(V(x)\wedge V(x-))+h$ and $V(v-)\geq(V(x)\wedge V(x-))+h$. Moreover, $x$ is an $h$-maximum for $V$ if $x$ is an $h$-minimum for $-V$, and $x$ is an $h$-extremum for $V$ if it is an $h$-maximum or an $h$-minimum for $V$.

Since $V$ is not a compound Poisson process, it is known (see Proposition VI.4, in [7]) that it takes pairwise distinct values in its local extrema. Combining this with the fact that $V$ has almost surely càd-làg paths and drifts to $-\infty$ without being the opposite of a subordinator, we can check that the set of $h$-extrema is discrete, forms a sequence indexed by $\mathbb{Z}$, unbounded from below and above, and that the $h$-minima and $h$-maxima alternate. Let $\mathcal{V}\subset D(\mathbb{R},\mathbb{R})$ be the set of the environments $v$ that satisfy the above properties and that are such that

[TABLE]

Note that the path of $V$ belongs to $\mathcal{V}$ with probability $1$.

We denote respectively by $(m_{i},\ i\in\mathbb{Z})$ and $(M_{i},\ i\in\mathbb{Z})$ the increasing sequences of $h$-minima and of $h$-maxima of $V$, such that $m_{0}\leq 0<m_{1}$ and $m_{i}<M_{i}<m_{i+1}$ for every $i\in\mathbb{Z}$. An $h$-valley is the fragment of the trajectory of $V$ between two $h$-maxima.

The valleys are visited successively by the diffusion. For the size of the valleys to be well adapted with respect to the time scale, we have to make the size of the valleys grow with time $t$. We are thus interested in $h_{t}$-valleys where

[TABLE]

where $\omega>1$ will be chosen later in accordance with some other parameters. We also define $N_{t}$, the index of the largest $h_{t}$-minima visited by $X$ until time $t$,

[TABLE]

We need deterministic bounds for the number of visited valleys. We define

[TABLE]

where $\delta>0$ is small enough so that $(1+3\delta)\kappa<1$, $\rho\in]0,\kappa/(1+\kappa)[$, and $\delta$ and $\rho$ are fixed once and for all in all the paper. The following lemma says that with high probability, $\tilde{n}_{t}\leq N_{t}\leq n_{t}$:

Lemma 2.1**.**

There is a positive constant $c$ such that for all $t$ large enough,

[TABLE]

We recall the definition of the standard valleys given in Section 3.3 of [46]. Their interest is mainly the fact that they are defined via successive stopping times, which make them convenient to use in the calculations, and also the fact that they take into consideration the descending phases between two $h_{t}$-minima.

Recall that $\delta>0$ is small enough so that $(1+3\delta)\kappa<1$. Assume $t$ is large enough so that $e^{(1-\delta)\kappa h_{t}}\geq h_{t}$. We define $\tilde{\tau}_{0}(h_{t})=\tilde{L}_{0}:=0$ and recursively for $i\geq 1$,

[TABLE]

These random variables depend on $h_{t}$ and therefore on $t$, even if this does not appear in the notations. We also define

[TABLE]

We call $i^{th}$ standard valley the re-centered truncated potential $(\tilde{V}^{(i)}(x),\ \tilde{L}_{i-1}\leq x\leq\tilde{L}_{i})$. The law of the bottom of these valleys is given in Fact 4.7 of Section 4.

Similarly as in [46] we define the deep bottoms of the $j^{th}$ standard valleys to be the interval

[TABLE]

Remark 2.2**.**

The random times $\tilde{L}_{i}^{\sharp}$, $\tilde{\tau}_{i}(h_{t})$, and $\tilde{L}_{i}$ are stopping times. As a consequence, the sequence $(\tilde{V}^{(i)}(x+\tilde{m}_{i}),\ \tilde{L}_{i-1}-\tilde{m}_{i}\leq x\leq\tilde{L}_{i}-\tilde{m}_{i})_{i\geq 1}$ is iid.

We also have that the sequence $(\tilde{m}_{i})_{i\geq 1}$ of the minima of the standard valleys coincides with the sequence $(m_{i})_{i\geq 1}$ with high probability for a large number of indices. Let

[TABLE]

Note from the definition of $\mathcal{V}$ that the sequences $(\tilde{m}_{i})_{i\geq 1}$ and $(m_{i})_{i\geq 1}$ are always defined for any $v\in\mathcal{V}$ so in particular the event $\mathcal{V}_{t}$ is well defined. We have:

Fact 2.3**.**

(Lemma 3.5 of [46])

There is a positive constant $c$ such that for all $t$ large enough,

[TABLE]

We define $X_{\tilde{m}_{j}}:=X(.+\tilde{m}_{j})$ which is, according to the Markov property, a diffusion in potential $V$ starting from $\tilde{m}_{j}$. We also define, for any $r\in\mathbb{R}$, $H_{X_{\tilde{m}_{j}}}(r)$ to be the hitting time of $r$ by $X_{\tilde{m}_{j}}$. When we deal with $X_{\tilde{m}_{j}}$ we often need the notation $A^{j}(x)=\int_{\tilde{m}_{j}}^{x}e^{\tilde{V}^{(j)}(s)}ds$.

As in [5] and [46], we approximate the distribution function of the renormalized local time by distribution functions of functionals of the sequence $(e_{j}S_{j}^{t},e_{j}S_{j}^{t}R_{j}^{t})_{j\geq 1}$ where

[TABLE]

In Section 4 of [46], it is shown that $e_{j}$ follows an exponential distribution with parameter $1/2$ (since the distribution of $e_{j}$ does not depend on $t$ we omit the dependence in $t$ for $e_{j}$ in the notations) and that the random variables $e_{j},S_{j}^{t},R_{j}^{t},\ j\geq 1$ are mutually independent. To simplify notations we define, as in [5] and [46], the process of the renormalized sum of the contributions:

[TABLE]

and the overshoots of $\sum_{i=1}^{.}e_{i}S_{i}^{t}R_{i}^{t}$: for any $a\geq 0$, let us define

[TABLE]

We have

Fact 2.4**.**

(Proposition 4.2 of [46])

$(Y_{1}^{t},Y_{2}^{t})$* converges in distribution in $(D([0,+\infty[,\mathbb{R}^{2}),J_{1})$ to a non-trivial bidimentional $\kappa$-stable subordinator $(\mathcal{Y}_{1},\mathcal{Y}_{2})$.*

Let us introduce some events that occur with high probability. They describe the behavior of the diffusion and provide effective approximations for the time and the local time, in the study of the case $0<\kappa<1$. On these events, the diffusion leaves the valleys from the right and never goes back to a previous valley, the local time and the time spent by the diffusion are negligible, compared with $t$, outside the bottoms of the valleys, the supremum of the local time and the time spent by the diffusion in the bottom of the valleys are approximated by the iid sequence $(e_{j}S_{j}^{t},e_{j}S_{j}^{t}R_{j}^{t})_{j\geq 1}$.

[TABLE]

Here, $\tilde{c}$ is a fixed positive constant that has been chosen small enough (the constraints for the choice of $\tilde{c}$ will be specified in the proofs of Fact 2.5 and Fact 2.7). Note that the above events depend both on the environment $V$ and on the Brownian motion driving the diffusion.

Fact 2.5**.**

There is a positive constant $L$ such that for all $t$ large enough,

[TABLE]

Fix $\eta\in]0,1[$. If $t$ is so large such that $(1-e^{-\tilde{c}h_{t}})^{-1}<(1+\eta)$ and $(1+e^{-\tilde{c}h_{t}})^{-1}(1-2/\log(h_{t}))\geq(1-\eta)$, then

[TABLE]

Our proofs are based on the study of the asymptotics of the quantities $\mathbb{P}(\mathcal{L}^{*}_{X}(t)\geq tx_{t})$ and $\mathbb{P}(\mathcal{L}^{*}_{X}(t)\leq t/x_{t})$ where $x_{t}$ depends on $t$ and goes to infinity with $t$. More precisely we define $x_{t}$ such that

[TABLE]

where $D>0$ and $\mu\in]1,2]$. Precise choices of $x_{t}$ will be made later, they will all satisfy (2.29) for some $D>0$ and $\mu\in]1,2]$.

Fix $\epsilon$ small enough so that Fact 4.8 of Section 4 is satisfied. We define $\mathcal{G}_{t}$ to be the set of "good environments" in the following way ; $v\in\mathcal{V}_{t}$ belongs to $\mathcal{G}_{t}$ if it satisfies the following conditions:

[TABLE]

where $P^{v}(.)$ is defined in Subsection 1.3 and $L$ is the constant defined in Fact 2.5. Note that, since $\mathcal{G}_{t}\subset\mathcal{V}_{t}$, we will often use the fact that $\tilde{m}_{j}=m_{j}$ for $v\in\mathcal{G}_{t}$ and $j\leq n_{t}$.

Lemma 2.6**.**

There is a positive constant $c$ such that for all $t$ large enough,

[TABLE]

We need that, as in Lemma 5.3 of [5], an inequality for the local time in the bottom of the $k^{th}$ valley is related to an inequality for the random variable $R_{k}^{t}$. Since we deal with unlikely inequalities for the local time, we have to prove the negligibility of the event where an inequality is satisfied for the local time, but not the corresponding inequality for $R_{k}^{t}$. Recall the notations $X_{\tilde{m}_{j}}$ and $H_{X_{\tilde{m}_{j}}}(.)$ defined above. For any fixed environment $v\in\mathcal{G}_{t}$, $k\in\{1,...,n_{t}\}$, and $z\in[0,1]$ we define

[TABLE]

where $\tilde{c}$ is the same as in the definitions of $\mathcal{E}^{4}_{t},\mathcal{E}^{6}_{t}$ and $\mathcal{E}^{7}_{t}$. Note that the inequalitiy $(1-z)(1+e^{-\tilde{c}h_{t}})/(1-e^{-\tilde{c}h_{t}})<x_{t}R_{k}^{t}$ (resp. $R_{k}^{t}/x_{t}<(1-e^{-\tilde{c}h_{t}})(1-z)$) only depends on the environment $v$. We consider $\mathcal{E}^{8}_{t}(v,k,z)$ (resp. $\mathcal{E}^{9}_{t}(v,k,z)$) to equal $\emptyset$ when $v$ is such that this equality is not satisfied. In the rest of the paper, we use the same convention when we consider events, at fixed environment, that partially depend on the environment.

Fact 2.7**.**

*There is a positive constant $c$ such that for $t$ large enough and any fixed environment $v\in\mathcal{G}_{t}$ we have *

[TABLE]

Note that in the above fact, $v$ is fixed in $\mathcal{G}_{t}\subset\mathcal{V}_{t}$ so $N_{t}\geq k$ implies $H(\tilde{m}_{k})/t\leq 1$. The quantities in the above fact are thus well defined.

We know from Fact 2.5 that the contributions to the local time and to the time spent of the successive valleys are approximated by the iid sequence $(e_{j}S_{j}^{t},e_{j}S_{j}^{t}R_{j}^{t})_{j\geq 1}$. Since we deal with extreme values of the local time, we need to know the right tails of the distributions of $e_{j}S_{j}^{t}$ and of $e_{j}S_{j}^{t}R_{j}^{t}$. We also need informations about the extreme values of $R_{j}^{t}$. Recall the definition of $\mathcal{R}$ in the beginning of Section 1.1. We have the following fact from [46]:

Fact 2.8**.**

Fix $\eta\in]0,1/3[$ and let $\mathcal{C}^{\prime}$ be the constant in Lemma 4.16 of [46]. We have

[TABLE]

$(R_{1}^{t})_{t>0}$* converges in distribution to $\mathcal{R}$ and there exists a positive $\lambda_{0}$ such that*

[TABLE]

where the above quantities are all finite. This entails the convergence of the moments of any positive order of $R_{1}^{t}$ to those of $\mathcal{R}$ when $t$ goes to infinity.

Finally, let us state a general lemma about the diffusion $X$:

Lemma 2.9**.**

Let $q$ be the constant in Theorem 1.4 of [46]. There are positive constants $c,K$ such that for all $r$ and $t$ large enough,

[TABLE]

2.2. Decomposition of the diffusion into independent parts

A necessary point for our proofs is to give a decomposition of the trajectory of $X$ that makes independence appear in order to apply the Borel-Cantelli Lemma. Let us fix $a>1$ and define the sequences $t_{n}:=e^{n^{a}}$, $u_{n}:=e^{\kappa(n^{a}-2an^{a-1}/3)}$, $v_{n}:=e^{\kappa(n^{a}-an^{a-1}/3)}$. Let $X^{n}:=X(H(v_{n})+.)$, the diffusion shifted by the hitting time of $v_{n}$ and $T_{n}:=\min\{t_{n},\tau(X^{n},u_{n}),\tau(X^{n},u_{n+1})\}$. Note that from the Markov property for $X$ at time $H(v_{n})$ and the stationarity of the increments of $V$, $X^{n}-v_{n}$ is equal in law to $X$ under the annealed probability $\mathbb{P}$. Let $n_{0}$ be large enough so that $u_{n}\leq v_{n}\leq u_{n+1}$ for all $n\geq n_{0}$. Clearly the sequence of processes $(X^{n}(t),\ 0\leq t\leq T_{n})_{n\geq n_{0}}$ is independent. We define the events

[TABLE]

The idea is that, since the sequence $(X^{n}(t),\ 0\leq t\leq T_{n})_{n\geq n_{0}}$ is independent, intersecting a sequence $(\mathcal{B}_{n})_{n\geq n_{0}}$ of interesting events (where each event $\mathcal{B}_{n}$ only depends on $(X^{n}(t),\ 0\leq t\leq t_{n})$) with $\mathcal{C}_{n}$ will result in $(\mathcal{B}_{n}\cap\mathcal{C}_{n})_{n\geq n_{0}}$, a sequence of independent events. Since $X^{n}$ is $X$ shifted by $H(v_{n})$, the event $\mathcal{D}_{n}$ is useful to neglect the time shift when dealing with the renormalization of the local time. We will need the following lemma:

Lemma 2.10**.**

[TABLE]

and

[TABLE]

Proof.

Here again, let $q$ be the constant in Theorem 1.4 of [46]. First, notice from the definitions of $t_{n}$, $u_{n}$ and $v_{n}$ that for all $n$ large enough,

[TABLE]

and

[TABLE]

From the definition of $\mathcal{C}_{n}$ and the Markov property applied to $X$ at time $H(v_{n})$ we have

[TABLE]

where $c$ is the constant in Lemma 2.9. The last inequality is true for $n$ large enough and comes, for the first term, from the combination of (2.48) and (2.42), and, for the second term, from (2.44). Recall that $e^{-h_{t_{n}}}\approx e^{-\log(t_{n})}=e^{-n^{a}}$. We thus deduce (2.46).

From the definition of $\mathcal{D}_{n}$, (2.49) and (2.43) we have for all $n$ large enough,

[TABLE]

where $c$ is the constant in Lemma 2.9. Since $e^{-c\phi(t_{n}/n)}=e^{-c(\log(\log(t_{n}/n)))^{\omega}}\approx e^{-ca^{\omega}(\log(n))^{\omega}}$ we deduce (2.47).

∎

2.3. The $\limsup$

We study the asymptotic of the quantity $\mathbb{P}(\mathcal{L}^{*}_{X}(t)/t\geq x_{t})$. Recall that $x_{t}$ is defined in (2.29) where $D>0$ and $\mu\in]1,2]$ are fixed constants. In all this subsection the parameter $\omega$ in (2.23) is fixed in $]1,\mu[$. We have:

Proposition 2.11**.**

There is a positive constant $c$ such that for all $a>1$ and $t$ large enough we have,

[TABLE]

Note that the functional of $(Y_{1}^{t},Y_{2}^{t})$ involved in this proposition is $Y_{1}^{\natural,t}(Y_{2}^{-1,t}(.)-)$, which represents the supremum of the local time before, and not including, the last valley (see the discussion after Theorem 1.1 of [46]). Even though, as we see in Proposition 4.1 of [46], the distribution function of $\mathcal{L}^{*}_{X}(t)/t$ involves a complex functional of $(Y_{1}^{t},Y_{2}^{t})$ that represents the last valley (together with the functional $Y_{1}^{\natural,t}(Y_{2}^{-1,t}(.)-)$ that represents the previous valleys), Proposition 2.11 says that the right tail of this distribution function does not involves the last valley. Before proving this proposition we prove some lemmas.

Lemma 2.12**.**

There is a positive constant $C$ such that for $t$ large enough,

[TABLE]

Proof.

Since for all $k\geq 1$ we have almost surely $H(\tilde{m}_{k})\geq\sum_{j=1}^{k-1}H(\tilde{L}_{j})-H(\tilde{m}_{j})$, we deduce that

[TABLE]

In the above we have used the definition of $\mathcal{E}^{7}_{t}$, Markov’s inequality, (2.27), the fact that the sequence $(e_{j}S_{j}^{t}R_{j}^{t})_{j\geq 1}$ is iid and the definition of $n_{t}$. For $t$ large enough we thus get

[TABLE]

Then,

[TABLE]

We can now use (2.40) with $\eta=1/6$ and we get that for all $t$ large enough

[TABLE]

Putting into (2.50), we get the result for $t$ large enough.

∎

We now link the asymptotic of $\mathbb{P}(R_{1}^{t}\leq a/x_{t})$ with the left tail of $I(V^{\uparrow})$. We have to make a distinction between the case where $V$ possesses negative jumps and the case where $V$ possesses no negative jumps, that is, $V$ is the $\kappa$-drifted Brownian motion. $R_{1}^{t}$ is an exponential functional of the bottom of the first valley. In the first case, due to the jumps, the left side of the bottom of the valley can be neglected, so only the right side matters. In the second case, both sides have the same law, so both have to be taken in consideration in the left tail of $R_{1}^{t}$.

Lemma 2.13**.**

Let $z_{t}$ go to infinity with $t$ and satisfying $(\log(z_{t}))^{2}<<h_{t}$.

• •

Assume that $V$ possesses negative jumps, then there is a positive constant $c$ such that for any $a>1$ and $t$ large enough,

[TABLE]

• •

Assume now that $V=W_{\kappa}$, the $\kappa$-drifted Brownian motion, then for $t$ large enough,

[TABLE]

where $\mathcal{R}$ is defined in the beginning of Section 1.1.

Proof.

We first assume that $V$ possesses negative jumps. Recall that $R_{1}^{t}=\int_{\tilde{\tau}_{1}^{-}(h_{t}/2)}^{\tilde{\tau}_{1}^{+}(h_{t}/2)}e^{-\tilde{V}^{(1)}(u)}\textnormal{d}u$ so, using the equality in law between $(\tilde{V}^{(i)}(\tilde{m}_{i}+x),\ 0\leq x\leq\tilde{\tau}_{i}(h)-\tilde{m}_{i})$ and $(V^{\uparrow}(x),\ 0\leq x\leq\tau(V^{\uparrow},h))$ given by Fact 4.7, we get

[TABLE]

so

[TABLE]

According to Lemma 4.6, for some positive constant $c$ and $t$ large enough, $(1-e^{-ch_{t}})\times\mathbb{P}\left(\int_{0}^{\tau(V^{\uparrow},h_{t}/2)}e^{-V^{\uparrow}(u)}du\leq 1/z_{t}\right)$ is less than

[TABLE]

Combining with (2.53) we get that for $t$ large enough,

[TABLE]

because $e^{-h_{t}/4}\leq(a-1)/z_{t}$ for large $t$ (thanks to the hypothesis $(\log(z_{t}))^{2}<<h_{t}$). This is the asserted upper bound in (2.51).

On the other hand, using the independence between the left and right parts of the valleys (given by Fact 4.7), we get

[TABLE]

From Fact 4.7, we get that the first factor equals

[TABLE]

while the second factor is more than

[TABLE]

Then, $c_{h_{t}}\geq c_{1}$ when $h_{t}\geq 1$ so, putting in (2.55), we get that for $t$ large enough,

[TABLE]

According to the combination of the first point of Theorem 1.15 of [45] with the second point of Theorem 1.16 of [45], there is a positive constant $c$ such that for $t$ large enough,

[TABLE]

Thanks to the hypothesis $(\log(z_{t}))^{2}<<h_{t}$ we deduce that, for $c$ decreased a little, the second factor in the right hand side of (2.56) is more than $e^{-c\left(\log((1-1/a)/z_{t})\right)^{2}}$. This yields the lower bound in (2.51).

We now consider the case where $V$ is the $\kappa$-drifted Brownian motion $W_{\kappa}$. Let $Z_{1}$ and $Z_{2}$ be two independent versions of the process $W_{\kappa}^{\uparrow}$. Since $W_{\kappa}$ has no jumps, the density of the process $P^{(2)}$ in Fact 4.7 is almost surely constant so $P^{(2)}$ is equal in law to $(\hat{W_{\kappa}}^{\uparrow}(x),\ 0\leq x\leq\tau(\hat{W_{\kappa}}^{\uparrow},h_{t}+))=(W_{\kappa}^{\uparrow}(x),\ 0\leq x\leq\tau(W_{\kappa}^{\uparrow},h_{t}))$ (the last equality comes from the fact that $\hat{W_{\kappa}}^{\uparrow}=W_{\kappa}^{\uparrow}$ and $W_{\kappa}$ is continuous). Combining this with (4.126) and the equality in law between $(\tilde{V}^{(i)}(\tilde{m}_{i}+x),\ 0\leq x\leq\tilde{\tau}_{i}(h)-\tilde{m}_{i})$ and $(V^{\uparrow}(x),\ 0\leq x\leq\tau(V^{\uparrow},h))$ (both are from Fact 4.7), we get

[TABLE]

Reasoning as in (2.54) we get for some positive constant $c$:

[TABLE]

Combining with (2.57) we get that for $t$ large enough,

[TABLE]

because $2e^{-h_{t}/4}\leq(a-1)/z_{t}$ for large $t$ and because, form the definitions of $\mathcal{R}$, $Z_{1}$ and $Z_{2}$ we have $\mathcal{R}\overset{\mathcal{L}}{=}I(Z_{1})+I(Z_{2})$. The above inequality is the asserted upper bound in (2.52).

On the other hand, we have trivially

[TABLE]

and combining with (2.58) we get that for $t$ large enough,

[TABLE]

This yields the lower bound in (2.52).

∎

The next lemma is crucial for our analysis and shows that, surprisingly, the contribution of the last valley can be neglected when we deal with the extremely large values of the local time, even though the contribution of the last valley appears in the limit distribution of $\mathcal{L}^{*}_{X}(t)/t$. The removing of the last valley allowed by this lemma is a key point in proving (in Proposition 2.11) that $\mathbb{P}(\mathcal{L}^{*}_{X}(t)/t\geq x_{t})$ is very closely related to simpler quantities of the type of $\mathbb{P}(Y_{1}^{\natural,t}(Y_{2}^{-1,t}(1)-)\geq x_{t})$, for which we will be able to describe the precise asymptotic (in Proposition 2.15). Therefore, having this lemma is the reason why we get precise results for the $\limsup$, and unfortunately a counterpart does not hold for the $\liminf$, as we will see in Section 2.4, which is why we get slightly less precise results for the $\liminf$. Recall the definition of $\mathcal{D}_{j}$ in (2.26). We have:

Lemma 2.14**.**

There are positive constants $C$ and $c$ such that for all $u>1$ and $t$ large enough,

[TABLE]

Proof.

We fix $v\in\mathcal{V}$, a realization of the environment. Let us define

[TABLE]

We have

[TABLE]

The fact that the sum stops at $n_{t}$ comes from $N_{t}<n_{t}$ together with the fact that $v\in\mathcal{G}_{t}\subset\mathcal{V}_{t}$. From the definitions of $\mathcal{E}_{t}(v,k,z)$, $\mathcal{E}^{8}_{t}(v,k,z)$, $\mathcal{E}^{5}_{t}$ and $\mathcal{E}^{7}_{t}$ we have, for $v\in\mathcal{G}_{t}$,

[TABLE]

When $z\leq 1-4/\log(h_{t})$ we have, on the big event in the right hand side,

[TABLE]

for $t$ large enough. Moreover, $(1-z)(1+e^{-\tilde{c}h_{t}})(1-e^{-\tilde{c}h_{t}})^{-1}\leq 1+e^{-\tilde{c}h_{t}/2}$ for $t$ large enough. As a consequence, for $z\in[0,1-4/\log(h_{t})]$ and $t$ large enough,

[TABLE]

Note also that the sum over $k$ in (2.59) corresponds to disjoint events (so it is actually the probability of a union of events). We thus get that $\mathds{1}_{v\in\mathcal{G}_{t}}\sum_{k=1}^{n_{t}}\int_{0}^{1-4/\log(h_{t})}P^{v}(\mathcal{E}_{t}(v,k,z),\ H(\tilde{m}_{k})/t\in dz)$ is less than

[TABLE]

Now, recall that $(S_{k}^{t},R_{k}^{t})$ only depends on $v$ and that, $v$ being fixed, $e_{k}$ belongs to the $\sigma$-field $\sigma\left(X(t),t\geq H(\tilde{m}_{k})\right)$. In other words, it only depends on the diffusion after time $H(\tilde{m}_{k})$. On the other hand, $H(\tilde{m}_{k})$ is measurable with respect to the $\sigma$-field $\sigma\left(X(t),0\leq t\leq H(\tilde{m}_{k})\right)$. From the Markov property applied to $X$ at $H(\tilde{m}_{k})$, we get that $H(\tilde{m}_{k})$ is independent from $(e_{k},S_{k}^{t},R_{k}^{t})$ so the above is less than

[TABLE]

In the above, $c$ is a positive constant and we have used the fact that $\tilde{L}_{k-1}\leq\tilde{m}_{k}$ for the first term and the fact that $v\in\mathcal{G}_{t}$ together with (2.35) and (2.36) for the second term. In conclusion we get

[TABLE]

Now, note that the first factor in the above product only depends on $(\tilde{v}^{(k)}(x),\ \tilde{L}_{k-1}\leq x\leq\tilde{L}_{k})$ (that is, only on $v$ shifted at $\tilde{L}_{k-1}$) while the second factor depends on $(v(x),\ x\leq\tilde{L}_{k-1})$ (that is, on $v$ before $\tilde{L}_{k-1}$). $\tilde{L}_{k-1}$ is a stopping time for the Lévy process $V$ of which $v$ is a fixed possible path. As a consequence, when we integrate (2.60) with respect to $v$ over $D(\mathbb{R},\mathbb{R})$ equipped with the probability mesure $P$, we get that the two factors are independent so $E[\mathds{1}_{V\in\mathcal{G}_{t}}\sum_{k=1}^{n_{t}}\int_{0}^{1-4/\log(h_{t})}P^{V}(\mathcal{E}_{t}(V,k,z),\ H(\tilde{m}_{k})/t\in dz)]$, is less than

[TABLE]

For the equality we have used the independence between $R_{k}^{t}$ and $e_{k}S_{k}^{t}$. Since the sequence $(e_{k},S_{k}^{t},R_{k}^{t})_{k\geq 1}$ is iid, we deduce that $E[\mathds{1}_{V\in\mathcal{G}_{t}}\sum_{k=1}^{n_{t}}\int_{0}^{1-4/\log(h_{t})}P^{V}(\mathcal{E}_{t}(V,k,z),\ H(\tilde{m}_{k})/t\in dz)]$ is less than

[TABLE]

Using (2.39) and Lemma 2.12 to bound respectively the second and third factor, we get the existence of a positive constant $C$ such that for $t$ large enough,

[TABLE]

It remains to study $E[\mathds{1}_{V\in\mathcal{G}_{t}}\sum_{k=1}^{n_{t}}\int_{1-4/\log(h_{t})}^{1}P^{V}(\mathcal{E}_{t}(V,k,z),\ H(\tilde{m}_{k})/t\in dz)]$. For $v\in\mathcal{G}_{t}$ and $z\in[1-4/\log(h_{t}),1]$ we have, using the definitions of $\mathcal{E}_{t}(v,k,z)$ and $\mathcal{E}^{8}_{t}(v,k,z)$:

[TABLE]

where, in the last inclusion, we used the fact that $(1+e^{-\tilde{c}h_{t}})/(1-e^{-\tilde{c}h_{t}})\leq 2$ for large $t$. Recall that the sum over $k$ corresponds to disjoint events, we thus get that

$\mathds{1}_{v\in\mathcal{G}_{t}}\sum_{k=1}^{n_{t}}\int_{1-4/\log(h_{t})}^{1}P^{v}(\mathcal{E}_{t}(v,k,z),\ H(\tilde{m}_{k})/t\in dz)$ is less than

[TABLE]

In the above, for the second inequality we have used the fact that $R_{k}^{t}$ only depends on $v$, and for the third we have used the fact that $\tilde{L}_{k-1}\leq\tilde{m}_{k}$ for the first term and (2.37) for the second term. Here again, the first factor in the above product only depends on $(\tilde{v}^{(k)}(x),\ \tilde{L}_{k-1}\leq x\leq\tilde{L}_{k})$ while the second factor depends on $(v(x),\ x\leq\tilde{L}_{k-1})$. As a consequence, when we integrate the above inequality with respect to $v$ over $D(\mathbb{R},\mathbb{R})$ equipped with the probability mesure $P$, we get that the two factors are independent so $E[\mathds{1}_{V\in\mathcal{G}_{t}}\sum_{k=1}^{n_{t}}\int_{1-4/\log(h_{t})}^{1}P^{V}(\mathcal{E}_{t}(V,k,z),\ H(\tilde{m}_{k})/t\in dz)]$ is less than

[TABLE]

where we have used the fact that the sequence $(R_{k}^{t})_{k\geq 1}$ is iid. Note that, from the definitions of $x_{t}$ and $h_{t}$ in respectively (2.29) and (2.23), we have $x_{t}\log(h_{t})\sim K(\log(\log(t)))^{\mu}$. Therefore, using Lemmas 2.13 (with $z_{t}=x_{t}\log(h_{t})/8$, $a=2$) and 2.12 to bound respectively the first and second factor we get that for $t$ large enough the above is less than

[TABLE]

for some positive constants $C$ and $K^{\prime}$. The term $Ce^{\kappa\phi(t)-\delta\kappa h_{t}/3}$ appears when $V=W_{\kappa}$ (because of (2.52)) and it is not necessary otherwise, note that this term is ultimately less than $e^{-c\phi(t)}$. Combining with (1.8) we get

[TABLE]

where $K^{\prime\prime}$ is a positive constant. Since we have chosen $\omega\in]1,\mu[$ in (2.23) we have $(\log(\log(t)))^{\mu}>>\phi(t)$ so for $t$ large enough the above inequality yields

[TABLE]

where the constant $c$ has been suitably decreased. Now putting (2.61) and (2.62) into (2.59) we get for some constant $c$ and $t$ large enough:

[TABLE]

Using (2.24), (2.27), and Lemma 2.6 we get the result for a suitably chosen constant $c$ and $t$ large enough.

∎

Proof.

of Proposition 2.11

Upper bound

[TABLE]

The event $\{N_{t}<n_{t}\}\cap\mathcal{E}^{1}_{t}$ ensures that for $j\leq N_{t}-1<n_{t}$, $\tilde{m}_{j}$ is no longer reached after $H(\tilde{L}_{j})$ and $\tilde{L}_{j}$ is no longer reached after $H(\tilde{m}_{j+1})$, and the event $\{V\in\mathcal{V}_{t},\ N_{t}<n_{t}\}$ ensures that $t$ is larger than $H(\tilde{m}_{N_{t}})\geq H(\tilde{L}_{N_{t}-1})$ and less than $H(\tilde{m}_{N_{t}+1})$. Therefore, for each $j\in\{1,...,N_{t}-1\}$ we have

[TABLE]

Moreover,

[TABLE]

The event $\{N_{t}<n_{t}\}\cap\mathcal{E}^{2}_{t}$ ensures that the local time does not grow too much between $H(\tilde{L}_{j-1})$ and $H(\tilde{m}_{j})$ for $j\in\{1,...,N_{t}+1\}$: $\sup_{y\in\mathbb{R}}(\mathcal{L}_{X}(H(\tilde{m}_{j}),y)-\mathcal{L}_{X}(H(\tilde{L}_{j-1}),y))\leq te^{(\kappa(1+3\delta)-1)\phi(t)}$. We thus see that, on $\{V\in\mathcal{V}_{t},\ N_{t}<n_{t}\}\cap\mathcal{E}^{1}_{t}\cap\mathcal{E}^{2}_{t}$, we have

[TABLE]

The term $\mathbb{P}(\mathcal{L}^{*}_{X}(t)\geq tx_{t},\ V\in\mathcal{V}_{t},\ N_{t}<n_{t},\ \mathcal{E}^{1}_{t},\mathcal{E}^{2}_{t})$ in the right hand side of (2.63) is therefore less than

[TABLE]

where $\tilde{x}_{t}^{1}:=x_{t}-e^{(\kappa(1+3\delta)-1)\phi(t)}\sim x_{t}$. Then, since $\tilde{x}_{t}^{1}$ converge to $+\infty$, we have $\tilde{x}_{t}^{1}\geq e^{-2\phi(t)}$ for $t$ large enough. Using the definition of $\mathcal{E}^{3}_{t}$, we get that for such large $t$ the above is less than

[TABLE]

where $\mathcal{D}_{j}$ is defined in (2.26). Putting all this together, we see that for $t$ large enough, $\mathbb{P}(\mathcal{L}^{*}_{X}(t)\geq tx_{t})$ is less than

[TABLE]

We now deal with the first term. Using the definition of $\mathcal{E}^{4}_{t}$ we get that the first term of (2.64) is less than

[TABLE]

where $\tilde{x}_{t}^{2}:=(1+e^{-\tilde{c}h_{t}})^{-1}\ \tilde{x}_{t}^{1}\sim x_{t}$. Now using the definition of $\mathcal{E}^{6}_{t}$, the above is less than

[TABLE]

where $\tilde{x}_{t}:=(1+e^{-\tilde{c}h_{t}})^{-1}\ \tilde{x}_{t}^{2}\sim x_{t}$. Using (2.28) with $\eta=a-1$, the above is less than

[TABLE]

Combining with (2.64) and the fact that $\tilde{x}_{t}^{1}\geq\tilde{x}_{t}$, we get that $\mathbb{P}(\mathcal{L}^{*}_{X}(t)\geq tx_{t})$ is less than

[TABLE]

Applying Lemma 2.14 with $u=\sqrt{a}$ (and $x_{t}$ replaced by $\tilde{x}_{t}$ which does not change anything since $\tilde{x}_{t}$ also satisfies (2.29)), Fact 2.3, (2.24) and (2.27), we deduce the existence of a positive constant $c$ such that for $t$ large enough,

[TABLE]

Since $\tilde{x}_{t}\sim x_{t}$ and $a>1$ we get the upper bound when $t$ is large enough.

Lower bound

[TABLE]

because, on $\{V\in\mathcal{V}_{t}\}\cap\{N_{t}<n_{t}\}\cap\mathcal{E}^{1}_{t}$, $\tilde{m}_{j}$ (for $j<N_{t})$ is no longer reached between times $H(\tilde{L}_{j})$ and $t$,

[TABLE]

because of the definition of $\mathcal{E}^{6}_{t}$ and where $\hat{x}_{t}:=(1-e^{-\tilde{c}h_{t}})^{-1}\ x_{t}\sim x_{t}$,

[TABLE]

where we used (2.28) with $\eta=1-a^{-1}$,

[TABLE]

Applying Fact 2.3, (2.24) and (2.27) we get the lower bound since $\hat{x}_{t}\sim x_{t}$ and $a>1$.

∎

Proposition 2.15**.**

Let $y_{t}$ go to infinity with $t$. For any $b>0$ and $u\in]0,1[$ there is a positive constant $C$ (depending on $b$ and $u$) such that for all $t$ large enough,

[TABLE]

Proof.

For any $z>0$, let $k^{t}(z)$ be the first index $k\geq 1$ such that $e_{k}S_{k}^{t}/t\geq z$. We have

[TABLE]

and $e_{k^{t}(y_{t})}S_{k^{t}(y_{t})}^{t}R_{k^{t}(y_{t})}^{t}$ has the same law as $e_{1}S_{1}^{t}R_{1}^{t}$ conditionally on $e_{1}S_{1}^{t}\geq ty_{t}$ so, using (2.65) and the independence between $e_{1}S_{1}^{t}$ and $R_{1}^{t}$:

[TABLE]

which proves the upper bound. For the lower bound, we fix $\eta<1-u$. Note that according to (2.65),

[TABLE]

and both events on the left-hand-side are independent so

[TABLE]

We first deal with the second factor. Let $(Z_{i})_{i\geq 1}$ be iid random variables such that $\mathcal{L}(Z_{1})=\mathcal{L}(e_{1}S_{1}^{t}R_{1}^{t}|e_{1}S_{1}^{t}\leq ty_{t})$ and $T$ be a geometric random variable with parameter $\mathbb{P}(e_{1}S_{1}^{t}\geq ty_{t})$, independent from the sequences $(Z_{i})_{i\geq 1}$ and $(e_{i}S_{i}^{t}R_{i}^{t})_{i\geq 1}$. We then have

[TABLE]

because the random variable $e_{i}S_{i}^{t}R_{i}^{t}$ is stochastically greater than the random variable $Z_{i}$. Then,

[TABLE]

On the first hand we have

[TABLE]

where we used Fact 2.4 and where $\mathcal{Y}_{2}$ is as in there, the second component of the limit process $(\mathcal{Y}_{1},\mathcal{Y}_{2})$. On the second hand

[TABLE]

Using (2.39) we get $\mathbb{P}\left(T\leq e^{\kappa\phi(t)}\right)\underset{t\rightarrow+\infty}{\sim}\mathcal{C}^{\prime}/y_{t}^{\kappa}$, where $\mathcal{C}^{\prime}$ is the constant in Fact 2.8. Putting all this together, we get the existence of a positive constant $c_{1}>0$ such that for $t$ large enough,

[TABLE]

We now study the first factor in the right hand side of (2.66). From the independence of the two factors $e_{1}S_{1}^{t}$ and $R_{1}^{t}$ in $e_{1}S_{1}^{t}R_{1}^{t}$ we have

[TABLE]

and

[TABLE]

where the limit comes from (2.39), because $y_{t}$ goes to infinity. We thus get the existence of a positive constant $c_{2}>0$ such that for $t$ large enough,

[TABLE]

Putting (2.67) and (2.68) in (2.66) we get the lower bound.

∎

Fix $\theta>1$. We apply Proposition 2.11 (the upper bound with $a=\theta^{1/3}$ and the lower bound with $a=\theta^{1/4}$), Proposition 2.15 (the upper bound applied with $b=\theta^{1/3}$, $y_{t}=\theta^{-1/3}x_{t}$ and the lower bound with $b=\theta^{-1/4}$, $y_{t}=\theta^{1/4}x_{t}$, $u=\theta^{-1/4}$) and Lemma 2.13 (the upper bounds of (2.51) and (2.52) applied with $a=\theta^{1/3}$, $z_{t}=\theta^{-2/3}x_{t}$, and the lower bounds of these same expressions applied with $a=\theta^{1/4}$, $z_{t}=\theta^{3/4}x_{t}$). We get:

Proposition 2.16**.**

Fix $\theta>1$. If $V$ possesses negative jumps, there are positive constants $\tilde{C}$ and $c$ such that for $t$ large enough,

[TABLE]

If $V:=W_{\kappa}$, the $\kappa$-drifted Brownian motion, there are positive constants $C$ and $c$ such that for $t$ large enough,

[TABLE]

We can now link the asymptotic behavior of the local time with the left tail of $I(V^{\uparrow})$:

Proof.

of Theorem 1.2

First, we assume that $V$ possesses negative jumps.

Let us assume that (1.13) is satisfied with some constants $\gamma>1$ and $C>0$. We now prove (1.14). Let $a>1$ and define the events

[TABLE]

We define $x_{t}:=C^{1-\gamma}a^{2}(\log(\log(t/a)))^{\gamma-1}$. Note that such a choice of $x_{t}$ satisfies (2.29) with $\mu=\gamma$ and $D=C^{1-\gamma}$. From the increase of $\mathcal{L}^{*}_{X}(.)$, (2.69) (the upper bound applied with $t=a^{n+1}$, $\theta=a$) and (1.13) we have, for $n$ large enough,

[TABLE]

Since $e^{-c\phi(a^{n+1})}=e^{-c(\log\log(a^{n+1}))^{\omega}}\leq n^{-2}$ for $n$ large enough, the above is the general term of a converging series so, using the Borel-Cantelli lemma, we deduce that $\mathbb{P}$-almost surely,

[TABLE]

and letting $a$ go to $1$ we get (1.14).

Now, let us assume that (1.15) is satisfied with some constants $\gamma>1$ and $C>0$. We now prove (1.16). Let $a>1$ and let $t_{n}$, $u_{n}$, $v_{n}$ (defined from this $a$) and $X^{n}$ be as in Subsection 2.2 (recall that $X^{n}-v_{n}$ is equal in law to $X$ under the annealed probability $\mathbb{P}$). We define

[TABLE]

We also define $\mathcal{C}_{n}$ and $\mathcal{D}_{n}$ to be as in Lemma 2.10 and $\mathcal{E}_{n}:=\mathcal{B}_{n}\cap\mathcal{C}_{n}$. We define $x_{t}:=C^{1-\gamma}(\log(\log(t)))^{\gamma-1}/a^{3(\gamma-1)}$. Note that such a choice of $x_{t}$ satisfies (2.29) with $\mu=\gamma$ and $D=C^{1-\gamma}/a^{3(\gamma-1)}$. According to (2.69) (the lower bound applied with $t=t_{n}$, $\theta=a^{\gamma-1}$), the definition of $t_{n}$, (1.15) and the fact that $n$ is large we have, for some positive constants $c_{a}$, $K_{a}$ and $n$ large enough,

[TABLE]

Since $e^{-c\phi(t_{n})}=e^{-c(\log\log(t_{n}))^{\omega}}\leq n^{-2}$ for $n$ large enough, we get

[TABLE]

Then, the combination of (2.71) and (2.46) yields

[TABLE]

Note that each event $\mathcal{E}_{n}$ belongs to the $\sigma$-field $\sigma(V(s)-V(u_{n}),u_{n}\leq s\leq u_{n+1},\ X(t),H(v_{n})\leq t\leq H(v_{n})+T_{n})$, in other words, it only depends on the diffusion between times $H(v_{n})$ and $H(v_{n})+T_{n}$ and on the environment between positions $u_{n}$ and $u_{n+1}$. From the Markov property and the independence of the increments of the environment, we get that the events $(\mathcal{E}_{n})_{n\geq 1}$ are independent. Combining this independence with (2.72) and the Borel-Cantelli Lemma we get that $\mathbb{P}$-almost surely, the event $\mathcal{E}_{n}$ is realized infinitely many often. For $n$ such that this event is realized we have

[TABLE]

According to (2.47) ans the Borel-Cantelli Lemma we have $\mathbb{P}$-almost surely

[TABLE]

so combining with (2.73) we deduce that $\mathbb{P}$-almost surely,

[TABLE]

and letting $a$ go to $1$ we get (1.16).

If $V=W_{\kappa}$, the $\kappa$-drifted Brownian motion with $0<\kappa<1$, we proceed the same proof, only replacing $I(V^{\uparrow})$ by $\mathcal{R}$ and (2.69) by (2.70). We thus get that the same result is stil true for $V=W_{\kappa}$, but with $\mathcal{R}$ instead of $I(V^{\uparrow})$.

∎

The other theorems for the $\limsup$ are now easy to prove.

Proof.

of Theorems 1.4 and 1.5

In the case where $V$ possesses negative jumps, Theorem 1.4 is a direct consequence of the combination of Theorem 1.2, (1.9) and (1.10). Similarly, the first point of Theorem 1.5 is obtained from the combination of Theorem 1.2 and (1.11). The second point of Theorem 1.5 is obtained from the combination of Theorem 1.2 and (1.12).

In the case where $V=W_{\kappa}$, the $\kappa$-drifted Brownian motion with $0<\kappa<1$, we only need to prove the last point of Theorem 1.5, and this requires to determine exactly the left tail of $\mathcal{R}$. This variable is equal in law to the sum of two independent random variables having the same law as $I(W_{k}^{\uparrow})$. We thus have

[TABLE]

where the equivalent comes from $(1.13)$ of [45]. Using this together with De Bruijn’s Theorem (see Theorem 4.12.9 in [8]) we get

[TABLE]

The last point of Theorem 1.5 follows from this combined with Theorem 1.2. ∎

2.4. The $\liminf$

For the $\liminf$ we study the asymptotic of the quantity $\mathbb{P}(\mathcal{L}^{*}_{X}(t)/t\leq 1/x_{t})$. Recall that $x_{t}$ is defined in (2.29) where $D>0$ and $\mu\in]1,2]$ are fixed constants. In all this subsection we take $\omega:=2$ for the parameter in (2.23).

Proposition 2.17**.**

Recall the $\lambda_{0}$ defined in Fact 2.8. There is a positive constant $c$ such that for all $a>1$ and $t$ large enough we have,

[TABLE]

Here, the functional of $(Y_{1}^{t},Y_{2}^{t})$ involved is $Y_{1}^{\natural,t}(Y_{2}^{-1,t}(.))$ which represents the supremum of the local time after leaving the last valley.

Proof.

Lower bound

For the lower bound we first have to prove that, if the local time at the bottom of the valleys is small, then the overall supremum of the local time is small. For this our argument starts similarly as in the proof of the lower bound of Proposition 4.1 in [46], by using the localization in the bottom of the valleys of the main contributions of the local time.

From the definition of $N_{t}$, we have $H(\tilde{m}_{N_{t}+1})\geq t$ on $\{V\in\mathcal{V}_{t},\ N_{t}<n_{t}\}$ so

[TABLE]

The event $\{N_{t}<n_{t}\}\cap\mathcal{E}^{1}_{t}$ ensures that for $j\leq N_{t}<n_{t}$, $\tilde{m}_{j}$ is no longer reached after $H(\tilde{L}_{j})$ and $\tilde{L}_{j}$ is no longer reached after $H(\tilde{m}_{j+1})$ so that for each $j\in\{1,...,N_{t}\}$ we have

[TABLE]

Moreover,

[TABLE]

The event $\{N_{t}<n_{t}\}\cap\mathcal{E}^{2}_{t}$ ensures that the local time does not grow too much between $H(\tilde{L}_{j-1})$ and $H(\tilde{m}_{j})$ for $j\in\{1,...,N_{t}+1\}$: $\sup_{y\in\mathbb{R}}(\mathcal{L}_{X}(H(\tilde{m}_{j}),y)-\mathcal{L}_{X}(H(\tilde{L}_{j-1}),y))\leq te^{(\kappa(1+3\delta)-1)\phi(t)}$. We thus see that, on $\{V\in\mathcal{V}_{t},\ N_{t}<n_{t}\}\cap\mathcal{E}^{1}_{t}\cap\mathcal{E}^{2}_{t}$, we have

[TABLE]

The right hand side of (2.74) is therefore more than

[TABLE]

where $\tilde{x}_{t}^{1}:=1/((1/x_{t})-e^{(\kappa(1+3\delta)-1)\phi(t)})$. Then, since $\tilde{x}_{t}^{1}\sim x_{t}$, we have $1/\tilde{x}_{t}^{1}\geq e^{-2\phi(t)}$ for $t$ large enough. Using the definition of $\mathcal{E}^{3}_{t}$, we get that for such large $t$ the above is more than

[TABLE]

where $\mathcal{D}_{j}$ is defined in (2.26). From the definition of $\mathcal{E}^{4}_{t}$ the above is more than

[TABLE]

where $\tilde{x}_{t}^{2}:=(1+e^{-\tilde{c}h_{t}})\tilde{x}_{t}^{1}\sim x_{t}$. Now using the definition of $\mathcal{E}^{6}_{t}$, the above is more than

[TABLE]

where $\tilde{x}_{t}:=(1+e^{-\tilde{c}h_{t}})\tilde{x}_{t}^{2}\sim x_{t}$. Let $a>1$. Using (2.28) with $\eta=a-1$, the above is more than

[TABLE]

where we used the definition of $(Y_{1}^{t},Y_{2}^{t})$. Applying Fact 2.3, (2.24) and (2.27) we get the asserted lower bound for a suitably chosen constant $c$ and $t$ large enough, since $\tilde{x}_{t}\sim x_{t}$ and $a>1$.

Upper bound

[TABLE]

because, on $\{V\in\mathcal{V}_{t},\ N_{t}<n_{t}\}\cap\mathcal{E}^{1}_{t}$, $\tilde{m}_{j}$ (for $j<N_{t})$ is no longer reached between times $H(\tilde{L}_{j})$ and $t$. We fix $v\in\mathcal{G}_{t}$, a realization of the environment. Let us define

[TABLE]

and $\nu_{t}(v,k,z):=P^{v}(\mathcal{E}_{t}(v,k,z))$. The event $\mathcal{E}_{t}(v,k,z)$ belongs to the $\sigma$-field $\sigma\left(X(t),t\geq H(\tilde{m}_{k})\right)$. In other words, it only depends on the diffusion after time $H(\tilde{m}_{k})$. On the other hand, $H(\tilde{m}_{k})$ and $\sup_{1\leq j\leq k-1}\mathcal{L}_{X}(H(\tilde{L}_{j}),\tilde{m}_{j})$ are measurable with respect to the $\sigma$-field $\sigma\left(X(t),0\leq t\leq H(\tilde{m}_{k})\right)$. From the Markov property applied to $X$ at $H(\tilde{m}_{k})$, we get that $H(\tilde{m}_{k})$ is independent from the event $\mathcal{E}_{t}(v,k,z)$. As a consequence, $P^{v}(\mathcal{L}_{X}(t,\tilde{m}_{N_{t}})\leq t/x_{t},\ \sup_{1\leq j\leq N_{t}-1}\mathcal{L}_{X}(H(\tilde{L}_{j}),\tilde{m}_{j})\leq t/x_{t},\ N_{t}<n_{t},\ \mathcal{E}^{1}_{t})$ is less than

[TABLE]

The fact that the sum stops at $n_{t}$ comes from $N_{t}<n_{t}$ together with the fact that $v\in\mathcal{G}_{t}\subset\mathcal{V}_{t}$. From the definition of $\mathcal{E}^{9}_{t}(v,k,z)$ we get

[TABLE]

We deduce that (2.76) is less than

[TABLE]

$v$ being fixed, the events in the right hand side of (2.77) are independent from the $\sigma$-field $\sigma(X(t),0\leq t\leq H(\tilde{m}_{k}))$. Using this independence and the fact that the sums over $k$ above corresponds to disjoint events (so they can be written as probabilites of unions of events), and integrating with respect to $v\in D(\mathbb{R},\mathbb{R})$ equipped with the probability measure $P$, we get that the first term in the right hand side of (2.75) is less than

[TABLE]

where $c$ is a positive constant and where we have used (2.38) for the second term, Lemma 2.6 for the third term, and the fact that $t$ is large enough,

[TABLE]

where we used the definitions of $\mathcal{E}^{5}_{t}$, $\mathcal{E}^{6}_{t}$ and $\mathcal{E}^{7}_{t}$,

[TABLE]

where we used the fact that $t$ is large enough, Fact 2.3, (2.24) and (2.27), and where the constant $c$ has been suitably decreased,

[TABLE]

On the event in the probability of the first term, we have $N_{t}=k^{t}(2/x_{t})$. For the second term we use (2.28) with $\eta=3/4$. The above is thus less than

[TABLE]

where, for the last inequality, we used the fact that the sequence $(R_{j}^{t})_{j\geq 1}$ is iid and independent from the random index $k^{t}(2/x_{t})$, (2.87) with $b=1/4$, $y_{t}=x_{t}/2$, and the definition of $(Y_{1}^{t},Y_{2}^{t})$. Then, note that according to (2.41) we have for all $t$ large enough,

[TABLE]

Bounding the three terms in the right hand side of (2.75) thanks to the above, Fact 2.3, (2.24) and (2.27) we get the upper bound for a suitably chosen constant $c$ and $t$ large enough.

∎

We now study the functional involved in Proposition 2.17. For this we need two lemmas.

In the remaining part of this subsection we fix $\eta\in]0,1/3[$. Let $y_{t}$ go to infinity with $t$ satisfying $\log(y_{t})<<\phi(t)$. Let $p_{t}:=\mathbb{P}(e_{1}S_{1}^{t}>t/y_{t})$ and $(\overline{\mathcal{H}_{i}})_{i\geq 1}$ be iid random variables such that $\overline{\mathcal{H}_{1}}$ has the same law as $e_{1}S_{1}^{t}R_{1}^{t}$ conditionally to $\{e_{1}S_{1}^{t}\leq t/y_{t}\}$: $\mathcal{L}(\overline{\mathcal{H}_{1}})=\mathcal{L}(e_{1}S_{1}^{t}R_{1}^{t}|e_{1}S_{1}^{t}\leq t/y_{t})$. Since we have $\log(y_{t})<<\phi(t)$, (2.39) gives $p_{t}\sim\mathcal{C}^{\prime}e^{-\kappa\phi(t)}y_{t}^{\kappa}$. We have

Lemma 2.18**.**

[TABLE]

where $\mathcal{C}^{\prime}$ is the constant in Fact 2.8 and $\mathcal{R}$ is defined in the beginning of Section 1.1.

Proof.

For any $\lambda\geq 0$ we have

[TABLE]

where we used iteration by parts,

[TABLE]

where we used Fubini’s Theorem,

[TABLE]

We now study the second and third term in $\eqref{queueloilimdiscretbis3lp}$. Using the fact that the difference between two points of a continuously differentiable function is the integral of its derivative, the last part of Fact 2.8 and the equivalent for $p_{t}$ we get

[TABLE]

where $\mathcal{C}^{\prime}$ is the constant in Fact 2.8. Then, from the last part of Fact 2.8 again,

[TABLE]

Recall that $\eta\in]0,1/3[$. $\int_{0}^{1/y_{t}}\mathbb{P}(e_{1}S_{1}^{t}/t>x)dx$ equals

[TABLE]

Now, the absolute values of the first and fourth terms of (2.82) are respectively less than $e^{-(1-2\eta)\phi(t)}$ and $\mathcal{C}^{\prime}e^{-(2\eta\kappa+(1-2\eta))\phi(t)}/(1-\kappa)$. In particular, thanks to $2\eta\kappa+(1-2\eta)>\kappa$ (which is trivial) and $\log(y_{t})<<\phi(t)$, both are negligible with respect to $e^{-\kappa\phi(t)}/y_{t}^{1-\kappa}$. From (2.39) we also have

[TABLE]

and the third term of (2.82) equals $\mathcal{C}^{\prime}e^{-\kappa\phi(t)}/(1-\kappa)y_{t}^{1-\kappa}$. Combining with (2.81) we get

[TABLE]

Putting together (2.80) and (2.83) in (2.79) we obtain (2.78).

∎

Lemma 2.19**.**

Recall the $\lambda_{0}$ defined in Fact 2.8. For any $\lambda\in[0,\lambda_{0}[$ we have

[TABLE]

and this limit is positive for all $\lambda\in]0,\lambda_{0}[$.

Proof.

For any $\lambda\in[0,\lambda_{0}[$ we have $\lambda y_{t}e_{1}S_{1}^{t}/t\in[0,\lambda_{0}[$ on the event $\{e_{1}S_{1}^{t}/t\leq 1/y_{t}\}$, so

[TABLE]

where we used iteration by parts and made the change of variable $y=y_{t}x$,

[TABLE]

where we used Fubini’s Theorem,

[TABLE]

According to (2.41) the second term is equivalent to $p_{t}(1-\mathbb{E}[e^{\lambda\mathcal{R}}])$. We now study the third term in $\eqref{queueloilimdiscretbis3lpge}$. Recall that $\eta\in]0,1/3[$. This term equals

[TABLE]

In the above, $\mathcal{C}^{\prime}$ is the constant in Fact 2.8. Now, thanks to (2.41), the absolute values of the first and fourth terms of (2.86) are ultimately less than $2\lambda\mathbb{E}[\mathcal{R}]y_{t}e^{-(1-2\eta)\phi(t)}$ and

$2\lambda\mathcal{C}^{\prime}\mathbb{E}[\mathcal{R}]y_{t}e^{-(2\eta\kappa+(1-2\eta))\phi(t)}/(1-\kappa)$. In particular, thanks to $2\eta\kappa+(1-2\eta)>\kappa$ (which is trivial) and $\log(y_{t})<<\phi(t)$, both are negligible with respect to $e^{-\kappa\phi(t)}y_{t}^{\kappa}$. From (2.39) and (2.41) we also have

[TABLE]

and, thanks to (2.41), the third term of (2.86) is equivalent to $\lambda\mathcal{C}^{\prime}y_{t}^{\kappa}e^{-\kappa\phi(t)}\int_{0}^{1}y^{-\kappa}\mathbb{E}[\mathcal{R}e^{\lambda y\mathcal{R}}]dy$. We thus get

[TABLE]

Putting into (2.85) we obtain (2.84).

We justify the positivity of the limit as follows: we see that the right hand side of (2.84) is equivalent to $\lambda\kappa\mathbb{E}[\mathcal{R}]/(1-\kappa)$ when $\lambda$ goes to [math]. The limit in (2.84) is therefore positive for small $\lambda$. On the other hand, $(\mathbb{E}[e^{\lambda y_{t}\overline{\mathcal{H}_{1}}/t}]-1)/p_{t}$ increases with $\lambda$ so the limit in (2.84) is non-decreasing on $[0,\lambda_{0}[$. We thus get the positivity of the limit for all $\lambda\in]0,\lambda_{0}[$.

∎

We can now study the lower and upper bounds given by Proposition 2.17:

Proposition 2.20**.**

Let $y_{t}$ be chosen as before (that is, $y_{t}\rightarrow+\infty$ and $\log(y_{t})<<\phi(t)$). There is a positive constant $L$ (not depending on the choice of $y_{t}$), such that for any $b>0$, $u>1$ and $t$ large enough we have

[TABLE]

Proof.

Lower bound

Let us fix $\alpha\in]b(1-\kappa)/\kappa\mathbb{E}[\mathcal{R}],ub(1-\kappa)/\kappa\mathbb{E}[\mathcal{R}][$. For any $z>0$, $k^{t}(z)$ still denotes the first index $k\geq 1$ such that $e_{k}S_{k}^{t}/t\geq z$. We have

[TABLE]

Now, recall that $(\overline{\mathcal{H}_{i}})_{i\geq 1}$ are iid random variables such that $\mathcal{L}(\overline{\mathcal{H}_{1}})=\mathcal{L}(e_{1}S_{1}^{t}R_{1}^{t}|e_{1}S_{1}^{t}\leq t/y_{t})$ and let $T$ be a geometric random variable with parameter $p_{t}=\mathbb{P}(e_{1}S_{1}^{t}>t/y_{t})$, independent from the sequence $(\overline{\mathcal{H}_{i}})_{i\geq 1}$. Recall that $p_{t}\sim\mathcal{C}^{\prime}e^{-\kappa\phi(t)}y_{t}^{\kappa}$, where $\mathcal{C}^{\prime}$ is the constant in Fact 2.8. We have

[TABLE]

where we put $h(t):=y_{t}/p_{t}\sim e^{\kappa\phi(t)}y_{t}^{1-\kappa}/\mathcal{C}^{\prime}$. We give a lower bound for the two factors in the left hand side of (2.88). We first study the Laplace transform of the normalized sum of the second factor to prove its convergence to a constant number. For any $\lambda\geq 0$, we have

[TABLE]

According to Lemma 2.18, the exponent is equivalent to

[TABLE]

and since $h(t)\sim e^{\kappa\phi(t)}y_{t}^{1-\kappa}/\mathcal{C}^{\prime}$ we get

[TABLE]

so $\sum_{i=1}^{\lfloor\alpha h(t)\rfloor}\overline{\mathcal{H}_{i}}/t$ converges in probability to $\alpha\kappa\mathbb{E}[\mathcal{R}]/(1-\kappa)$ which yields

[TABLE]

since $\alpha>b(1-\kappa)/\kappa\mathbb{E}[\mathcal{R}]$.

We now study the first factors in the left hand side of (2.88). Since $T$ is geometric with parameter $p_{t}$, we have

[TABLE]

Now, since $h(t)\sim e^{\kappa\phi(t)}y_{t}^{1-\kappa}/\mathcal{C}^{\prime}$ and $p_{t}\sim\mathcal{C}^{\prime}e^{-\kappa\phi(t)}y_{t}^{\kappa}$ we get

[TABLE]

Now, putting (2.89) and (2.90) into (2.88), and combining the latter with (2.87), we get the result for $t$ large enough since $\alpha<ub(1-\kappa)/\kappa\mathbb{E}[\mathcal{R}]$.

Upper bound

Recall (2.87) and the definitions of $(\overline{\mathcal{H}_{i}})_{i\geq 1}$, $p_{t}$, $T$ and $h(t)$. Let us fix $\alpha>0$ that will be chosen latter. We have

[TABLE]

Let us choose $\lambda\in]0,\lambda_{0}[$ where $\lambda_{0}$ defined in Fact 2.8. The second term equals

[TABLE]

According to Lemma 2.19, the fact that $h(t)\sim e^{\kappa\phi(t)}y_{t}^{1-\kappa}/\mathcal{C}^{\prime}$ and $p_{t}\sim\mathcal{C}^{\prime}e^{-\kappa\phi(t)}y_{t}^{\kappa}$ we have

[TABLE]

Thanks to the positivity of the limit in Lemma 2.19 we can choose $\alpha$ such that $0<\alpha<b\lambda/2(1-\mathbb{E}[e^{\lambda\mathcal{R}}]+\lambda\int_{0}^{1}x^{-\kappa}\mathbb{E}[\mathcal{R}e^{\lambda x\mathcal{R}}]dx)$. We thus get for $t$ large enough

[TABLE]

Since $T$ is geometric with parameter $p_{t}$, we have

[TABLE]

where we used the equivalents for $h(t)$ and $p_{t}$ and where the last inequality holds for $t$ large enough.

Now, putting (2.92) and (2.93) into (2.91), and combining the latter with (2.87), we get the result for $t$ large enough.

∎

Fix $\theta>1$. We apply Proposition 2.17 with $a=\theta^{1/3}$ and Proposition 2.20 (the lower bound with $b=\theta^{1/3}$, $u=\theta^{1/3}$, $y_{t}=\theta^{1/3}x_{t}$ and the upper bound with $b=1/4$, $y_{t}=x_{t}/2$). We get:

Proposition 2.21**.**

Let $\tilde{L}\in]0,\min\{L/8,\lambda_{0}/8\}[$ where $L$ is the positive constant defined in Proposition 2.20 and $\lambda_{0}$ is defined in Fact 2.8. There is a positive constant $c$ such that for any $\theta>1$ and $t$ large enough we have

[TABLE]

We can now prove Theorem 1.9.

Proof.

of Theorem 1.9

Recall that $\tilde{L}$ is the positive constant defined in Proposition 2.21 and let $x_{t}:=4\log(\log(t))/2\tilde{L}$. Note that such a choice of $x_{t}$ satisfies (2.29) with $\mu=2$ and $D=4/2\tilde{L}$. We define the events

[TABLE]

From the increase of $\mathcal{L}^{*}_{X}(.)$ and Proposition 2.21 (the upper bound applied with $t=2^{n}$) we have for $n$ large enough,

[TABLE]

Since $e^{-c\phi(2^{n})}=e^{-c(\log\log(2^{n}))^{2}}\leq n^{-2}$ for $n$ large enough, the above is the general term of a converging series so, using the Borel-Cantelli lemma we deduce that $\mathbb{P}$-almost surely,

[TABLE]

so the $\liminf$ is positive.

We now prove the upper bound for the $\liminf$. Let $a>1$ and let $t_{n}$, $u_{n}$, $v_{n}$ (defined from this $a$) and $X^{n}$ be as in Subsection 2.2 (recall that $X^{n}-v_{n}$ is equal in law to $X$ under the annealed probability $\mathbb{P}$). We define

[TABLE]

We also define $\mathcal{C}_{n}$ and $\mathcal{D}_{n}$ to be as in Lemma 2.10 and $\mathcal{E}_{n}:=\mathcal{B}_{n}\cap\mathcal{C}_{n}$. We define $x_{t}:=\kappa\mathbb{E}[\mathcal{R}]\log(\log(t))/a^{2}(1-\kappa)$. Note that such a choice of $x_{t}$ satisfies (2.29) with $\mu=2$ and $D=\kappa\mathbb{E}[\mathcal{R}]/a^{2}(1-\kappa)$. Recall the notation $\mathcal{L}^{*,+}_{X}$ defined in Subsection 1.3. Let us choose $K$ as in Lemma 2.9, $\eta$ and $C$ as in Lemma 3.2 of the next section and $Q$ be as defined in the next section. According to (2.45) and Lemma 3.2 applied with $u=t_{n}/n\log(\log(t_{n}))$, $v=v_{n}$ we get for all $n$ large enough,

[TABLE]

From the definition of $t_{n}$, the fact that $n^{\kappa}v_{n}/t_{n}^{\kappa}=n^{\kappa}e^{-\kappa 2an^{a-1}/3}$ and $\log(\log(t_{n}))=a\log(n)$, we get

[TABLE]

According to Proposition 2.21 (the lower bound applied with $t=t_{n}$, $\theta=a$) and the definition of $t_{n}$ we have

[TABLE]

Since $e^{-c\phi(t_{n})}=e^{-c(\log\log(t_{n}))^{2}}\leq n^{-2}$ for $n$ large enough, we get

[TABLE]

Then, the combination of (2.95) and (2.46) yields

[TABLE]

As in the proof of Theorem 1.2, we see that each event $\mathcal{E}_{n}$ belongs to the $\sigma$-field $\sigma(V(s)-V(u_{n}),u_{n}\leq s\leq u_{n+1},\ X(t),H(v_{n})\leq t\leq H(v_{n})+T_{n})$ so the events $(\mathcal{E}_{n})_{n\geq 1}$ are independent. Combining this independence with (2.96) and the Borel-Cantelli Lemma we get that $\mathbb{P}$-almost surely, the event $\mathcal{E}_{n}$ is realized infinitely many often. Combining with (2.94) and the Borel-Cantelli Lemma we get that $\mathbb{P}$-almost surely, the event $\mathcal{B}_{n}^{\prime}\cap\mathcal{B}_{n}\cap\mathcal{C}_{n}$ is realized infinitely many often. For $n$ such that this event is realized we have

[TABLE]

Recall that according to (2.47) and the Borel-Cantelli Lemma we have $\mathbb{P}$-almost surely

[TABLE]

so combining with (2.97) we deduce that $\mathbb{P}$-almost surely,

[TABLE]

and letting $a$ go to $1$ we get the asserted upper bound for the $\liminf$.

∎

3. Almost sure behavior when $\kappa>1$

In this section we prove Theorems 1.10 and 1.12. Let us first recall some facts and notations from [40] and [46]. Our proof is based on the study of the so-called generalized Ornstein-Uhlenbeck process defined by

[TABLE]

where $R$ is a two-dimensional squared Bessel process independent from $V$. Let $L$ be the local time of $Z$ for the position $1$, $n$ the associated excursion measure, and $L^{-1}$ the right continuous inverse of $L$. We denote by $\xi$ a generic excursion. Let us denote by $Q$ the positive constant denoted by $n[\zeta]$ in Section 5 of [40]. Recall also the notations $K$ and $m$ defined in the Introduction ($m$ only appears in the context where $\kappa>1$, where it is properly defined). We have:

Fact 3.1**.**

There is $\eta>0$ and $r_{0}>0$ such that for all $r\geq r_{0}$ and $h>1$ we have

[TABLE]

[TABLE]

The first point is Lemma 2.3 of [46] while the second point is Proposition 5.1 of [40]. Note that Fact 3.1 is true for a general positive $\kappa$ and not only for $\kappa>1$.

Let us recall the link between $Z$ and the local time until the hitting times. The local time at point $x$ and within the hitting time $H(r)$ is given by:

[TABLE]

$\mathcal{M}_{1}(r)$ and $\mathcal{M}_{2}(r)$ denote respectively the supremum of the above expression for $x\in]-\infty,0[$ and $x\in[0,+\infty[$. The supremum of the local time until instant $H(r)$ can be written

[TABLE]

where

[TABLE]

The boundedness of $\mathcal{M}_{1}(r)$ is justified in (2.9) of [46].

We can now study the behavior of the local time at a hitting time. This allows to prove the following useful lemma.

Lemma 3.2**.**

There exist $\eta>0$, a positive constant $C$, $u_{0}>0$ and $v_{0}>0$ such that

[TABLE]

where $\mathcal{L}^{*,+}_{X}$ is as defined in Subsection 1.3.

Proof.

From the definition of $\mathcal{M}_{2}$ and (3.101) we have

[TABLE]

Now let us choose $\eta$ and $v_{0}$ such that (3.98) is true for all $r\geq v_{0}$ and $h>1$ with this $\eta$. We choose $C>Q2^{\kappa}\Gamma(\kappa)\kappa^{2}K$ and $u_{0}$ such that $n(\sup\xi>u)\leq Cu^{-\kappa}$ for all $u\geq u_{0}$. Such a $u_{0}$ exists thanks to (3.99). For $u\geq u_{0}$ and $v\geq v_{0}$ we have

[TABLE]

Putting into (3.102) we get the result.

∎

Remark 3.3**.**

Neither Lemma 3.2 nor its proof require the hypothesis that $\kappa>1$. The lemma is thus true whatever is the value of $\kappa$.

We need to study the supremum of the local time until a deterministic time. The following fact, from Lemma 2.1 of [46], says that we can replace a deterministic time by a hitting time when $\kappa>1$. **We now assume $\kappa>1$ until the end of this section. ** We have:

Fact 3.4**.**

For any $\alpha\in]\max\{3/4,1/\kappa\},1[$ their exists $\eta>0$ such that for $r$ large enough we have

[TABLE]

Let $\alpha\in]\max\{3/4,1/\kappa\},1[$ be fixed until the end of this section and $\eta>0$ be small enough so that both (3.98) and (3.103) are satisfied (with this $\alpha$).

3.1. The $\liminf$

In this subsection we prove Theorem 1.12. Let us define $J:=2(\Gamma(\kappa)\kappa^{2}K/m)^{1/\kappa}$, the expected $\liminf$. We begin to prove that

[TABLE]

Let $a>1$ and define the events

[TABLE]

From the increase of $\mathcal{L}^{*}_{X}(.)$, (3.103), (3.100), (3.101), and (3.98), we have

[TABLE]

According to the equivalent given by (3.99), the exponent in the above expression is, for $n$ large enough, less than $-a\log(\log(a^{n}))$, so for such large $n$,

[TABLE]

The other two terms in the right hand side of (3.105) are also general terms of converging series so we obtain,

[TABLE]

According to the Borel-Cantelli lemma we get

[TABLE]

in which we can let $a$ go to $1$ which yields (3.104). We now prove that

[TABLE]

Let us fix $a>0$, $u_{n}:=n^{2n}$, $v_{n}:=u_{n}/m+u_{n}^{\alpha}=n^{2n}/m+n^{2\alpha n}$ and $X^{n}:=X(H(2v_{n})+.)$, the diffusion shifted by the hitting time of $2v_{n}$. Note that from the Markov property for $X$ at time $H(2v_{n})$ and the stationarity of the increments of $V$, $X^{n}-2v_{n}$ is equal in law to $X$ under the annealed probability $\mathbb{P}$. We take $n$ so large such that $2v_{n}<v_{n+1}$ and define the events

[TABLE]

Recall that $\eta>$ has been fixed so that (3.98) is satisfied. For this $\eta$ and for $C>Q2^{\kappa}\Gamma(\kappa)\kappa^{2}K$, the inequality of Lemma 3.2 is true for $u$ and $v$ large enough. According to this lemma applied with $u=aJ(u_{n+1}/\log(\log(u_{n+1})))^{1/\kappa}$, $v=2v_{n}$ we get for all $n$ large enough,

[TABLE]

Since, for $n$ large enough, $v_{n}/u_{n+1}\leq 1/mn^{2}$ and $\log(\log(u_{n+1}))\sim\log(n)$ we can deduce that

[TABLE]

From the equality in law between $X^{n}-2v_{n}$ and $X$ under $\mathbb{P}$, , (3.100), (3.101), (2.45), (3.98)

[TABLE]

According to the equivalent given by (3.99) and the definitions of $u_{n}$ and $v_{n}$, the exponent in the above expression is equivalent to $(1+a)^{-\kappa}\log(\log(u_{n+1}))\sim(1+a)^{-\kappa}\log(n)$ so for $n$ large enough,

[TABLE]

and the remaining terms in the right hand side of (3.108) are the general terms of converging series. We thus get

[TABLE]

[TABLE]

where we used the equality in law between $X^{n}-2v_{n}$ and $X$ under $\mathbb{P}$. Combining with (2.44) applied with $r=v_{n}$ we get

[TABLE]

Then, the combination of (3.109) and (3.110) yields

[TABLE]

According to the definitions of the sequences $(u_{n})_{n\geq 1}$ and $(u_{n})_{n\geq 1}$, to (3.103), and to the increase of $\mathcal{L}^{*,+}_{X}$, we have, for $n$ large enough, $\mathbb{P}(\mathcal{F}_{n})\leq u_{n}^{-\eta}$, so

[TABLE]

Note that each event $\mathcal{E}_{n}$ belongs to the $\sigma$-field $\sigma(V(s)-V(v_{n}),v_{n}\leq s\leq v_{n+1},\ X(t),H(2v_{n})\leq t\leq\min(\tau(X^{n},v_{n}),\tau(X^{n},v_{n+1})))$, in other words, it only depends on the diffusion between times $H(2v_{n})$ and $\min(\tau(X^{n},v_{n}),\tau(X^{n},v_{n+1}))$ and on the environment between positions $v_{n}$ and $v_{n+1}$. From the Markov property and the independence of the increments of the environment, we get that the events $(\mathcal{E}_{n})_{n\geq 1}$ are independent. Combining this independence with (3.111) and the Borel-Cantelli Lemma we get that $\mathbb{P}$-almost surely, the event $\mathcal{E}_{n}$ is realized infinitely many often.

Combining (3.107), (3.112) and the Borel-Cantelli Lemma we get that $\mathbb{P}$-almost surely the event $\mathcal{B}_{n}\cap\mathcal{F}_{n}$ is realized for all large $n$. We deduce that $\mathbb{P}$-almost surely, the event $\mathcal{B}_{n}\cap\mathcal{C}_{n}\cap\mathcal{D}_{n}\cap\mathcal{F}_{n+1}$ is realized infinitely many often. Then, for $n$ such that this event is realized we have,

[TABLE]

so

[TABLE]

Now, letting $a$ go to [math] and combining with the finiteness of $\sup_{]-\infty,0[}\mathcal{L}_{X}(+\infty)$ (see (3.100) and (3.101)) we obtain (3.106) so Theorem 1.12 is proved.

3.2. The $\limsup$

In this subsection we prove Theorem 1.10. First, let us assume that $\int_{1}^{+\infty}\frac{(f(t))^{\kappa}}{t}dt<+\infty$ and prove that

[TABLE]

According to Remark 1.20 the condition $\int_{1}^{+\infty}\frac{(f(t))^{\kappa}}{t}dt<+\infty$ is equivalent to

[TABLE]

Let us fix $a>0$ and define the events

[TABLE]

From the increase of $\mathcal{L}^{*,+}_{X}(.)$, (3.103), (3.100), (3.101), and (3.98), we have

[TABLE]

According to (3.99), the first term in the right hand side is equivalent to

[TABLE]

which is the general term of a convergent series, according to (3.114). The two remaining terms are also the general terms of convergent series so we get

[TABLE]

and applying the Borel-Cantelli lemma we deduce

[TABLE]

Now, letting $a$ go to [math] and combining with the finiteness of $\sup_{]-\infty,0[}\mathcal{L}_{X}(+\infty)$ (see (3.100) and (3.101)) we obtain (3.113).

Let us now assume that $\int_{1}^{+\infty}\frac{(f(t))^{\kappa}}{t}dt=+\infty$ and prove that

[TABLE]

According to Remark 1.20 the condition $\int_{1}^{+\infty}\frac{(f(t))^{\kappa}}{t}dt=+\infty$ is equivalent to

[TABLE]

Let $M>0$, $u_{n}:=2^{n}/m-2^{\alpha n}$ and $X^{n}:=X(H(\sqrt{2}u_{n})+.)$, the diffusion shifted by the hitting time of $\sqrt{2}u_{n}$. Note that from the Markov property for $X$ at time $H(\sqrt{2}u_{n})$ and the stationarity of the increments of $V$, $X^{n}-\sqrt{2}u_{n}$ is equal in law to $X$ under the annealed probability $\mathbb{P}$. We take $n$ so large such that $\sqrt{2}u_{n}<u_{n+1}$ and define the events

[TABLE]

From the equality in law between $X^{n}-\sqrt{2}u_{n}$ and $X$ under $\mathbb{P}$, (3.100), (3.101), and (3.98), we have

[TABLE]

According to (3.99) and the definition of $u_{n}$, the exponent in the right hand side is equivalent to

[TABLE]

Since $f$ converges to [math] at infinity, the above is also an equivalent for the term $1-e^{-(...)}$ in the right hand side of (3.117). Then, combining with (3.116) and the fact that the other term is the general term of a covering series we get

[TABLE]

Reasoning as in the proof of (3.110) we can prove that

[TABLE]

so

[TABLE]

According to (3.103), and to the increase of $\mathcal{L}_{X}^{*}$, we also prove that

[TABLE]

The events $\mathcal{E}_{n}$ are independent since for each $n$, $\mathcal{E}_{n}$ belongs to the $\sigma$-field $\sigma(V(s)-V(u_{n}),u_{n}\leq s\leq u_{n+1},\ X(t),H(\sqrt{2}u_{n})\leq t\leq H(\sqrt{2}u_{n})+\min(\tau(X^{n},u_{n}),\tau(X^{n+1},u_{n+1}))$. Combining this independence with (3.120) and the Borel-Cantelli Lemma we get that $\mathbb{P}$-almost surely, the event $\mathcal{E}_{n}$ is realized infinitely many often. Combining with (3.121) and the Borel-Cantelli Lemma we get that $\mathbb{P}$-almost surely, the event $\mathcal{C}_{n}\cap\mathcal{D}_{n}\cap\mathcal{F}_{n+1}$ is realized infinitely many often. Then, for $n$ such that this event is realized we have,

[TABLE]

so

[TABLE]

Now, letting $M$ go to infinity we get (3.115) so Theorem 1.10 is proved.

4. Some lemmas

In this section, we justify some technical facts and lemmas for $V$, $V^{\uparrow}$ and the diffusion in $V$. Some of them are known or can be easily obtained from results of [46], and we give some details for their justification when it is necessary. Some of these facts are new, like the approximation of the contributions of the valleys to the traveled distance by an iid sequence.

4.1. Properties of $V$, $V^{\uparrow}$ and $\hat{V}^{\uparrow}$

Lemma 4.1**.**

(Lemma 5.5 of [46])

There are two positive constants $c_{1},c_{2}$ such that

[TABLE]

Lemma 4.2**.**

There are positive constants $c_{1},c_{2}$ such that,

[TABLE]

Proof.

Let us choose $\gamma\in]0,\kappa[$, we have

[TABLE]

where we used Markov’s inequality for the first term and the Markov property at time $t$ for the second term, together with the fact that the supremum of $V$ on $[0,+\infty[$ follows an exponential distribution with parameter $\kappa$. Since $E[e^{\gamma V(t)}]=e^{t\Psi_{V}(\gamma)}$ and $\Psi_{V}(\gamma)<0$ (because $0<\gamma<\kappa$), we get the result with $c_{1}:=2\gamma$ and $c_{2}:=-\Psi_{V}(\gamma)$.

∎

Lemma 4.3**.**

(Lemma 5.3 of [46])

There is a positive constant $\mathcal{C}$ such that

[TABLE]

We now state some Lemmas about $V^{\uparrow}$. First, we recall how $V^{\uparrow}$ and $\hat{V}^{\uparrow}$ are defined.

$V$ being spectrally negative, the Markov family $(V^{\uparrow}_{x},x\geq 0)$ may be defined by $h$-transform as in [7], Section VII.3. For any $x\geq 0$, the process $V^{\uparrow}_{x}$ must be seen as $V$ conditioned to stay positive and starting from $x$. We denote $V^{\uparrow}$ for the process $V^{\uparrow}_{0}$. It is known that $V^{\uparrow}_{x}$ converges in the Skorokhod space to $V^{\uparrow}$ when $x$ goes to [math]. Also, as well as $V$, $V^{\uparrow}$ has no positive jumps so it reaches every positive level continuously.

We now define $\hat{V}^{\uparrow}$, that is, $\hat{V}$ conditioned to stay positive. Since $V$ is spectrally negative and not the opposite of a subordinator, it is regular for $]0,+\infty[$ (see [7], Theorem VII.1), so $\hat{V}$ is for $]-\infty,0[$. Moreover, $\hat{V}$ drifts to $+\infty$. We can thus define the Markov family $(\hat{V}^{\uparrow}_{x},\ x\geq 0)$ as in Doney [18], Chapter 8. It can be seen from there that for any $x>0$ the process $\hat{V}^{\uparrow}_{x}$ is Markovian and has infinite life-time. If moreover $V$ has unbounded variation then $\hat{V}$ is regular for $]0,+\infty[$, and from Theorem 24 of [18], we have that $\hat{V}^{\uparrow}_{0}$, that we denote by $\hat{V}^{\uparrow}$, is well defined.

Here again, for any $x\geq 0$, the process $\hat{V}^{\uparrow}_{x}$ must be seen as $\hat{V}$ conditioned to stay positive and starting from $x$. Note that, since $\hat{V}$ converges almost surely to infinity, for $x>0$, $\hat{V}^{\uparrow}_{x}$ is only $\hat{V}_{x}$ conditioned in the usual way to remain positive.

Lemma 4.4**.**

(Lemma 5.8 of [46])

There are two positive constants $c_{1},c_{2}$ such that, for all $1<a<b$, we have

[TABLE]

Lemma 4.5**.**

There are positive constants $c_{3},c_{4},c_{5},c_{6}$ such that,

[TABLE]

Proof.

Let us fix $0\leq x<y$. From the Markov property applied at $\tau(V^{\uparrow},x)$, the hitting time of $x$ by $V^{\uparrow}$, we have

[TABLE]

so (4.122) follows from Lemma 5.7 of [46]. For the second point, we have

[TABLE]

so taking the complementary,

[TABLE]

where, for the first term, we used (4.122) with $x=z$, $y=2z$ and, for the second term, we used the Markov property at time $\tau(V^{\uparrow}_{z},2z)$. Combining with Lemma 4.4 applied with $a=z$, $b=2z$, we get (4.123).

∎

Lemma 4.6**.**

There is a positive constant $c$ such that for $t$ large enough,

[TABLE]

Proof.

[TABLE]

where we used the Markov property at time $\tau(V^{\uparrow},h_{t}/2)$ for the equality in law. We thus get

[TABLE]

where, for the first term, we applied (4.123) with $z=h_{t}/2$, $r=e^{h_{t}/8}/2$ and, for the second term, we applied Lemma 4.4 with $a=3h_{t}/8$, $b=h_{t}/2$.

Then, according to Corollary VII.19 of [7] we have

[TABLE]

and, according to Theorem 1.1 of [45], $I(V^{\uparrow})$ admits some finite exponential moments, so in particular it has finite expectation. We thus get

[TABLE]

The result follows from the combination of (4.124) and (4.125).

∎

The next fact gives the law of the bottom of the valleys in terms of the laws of $V^{\uparrow}$ and $\hat{V}^{\uparrow}$. It is a combination of Propositions 3.2 and 3.6 of [46].

Fact 4.7**.**

Assume $V$ has unbounded variation. For all $i\geq 1$ let

[TABLE]

For all $i\geq 1$ we have

[TABLE]

where $d_{VT}$ is the total variation distance. Moreover, the law of $P^{(2)}$ is absolutely continuous with respect to the law of the process $(\hat{V}^{\uparrow}(x),\ 0\leq x\leq\tau(\hat{V}^{\uparrow},h_{t}+))$ and has density $c_{h_{t}}/(1-e^{-\kappa\hat{V}^{\uparrow}(\tau(\hat{V}^{\uparrow},h_{t}+))})$ with respect to this law, where $c_{h_{t}}$ is a constant increasing with $h_{t}$ and converging to $1$ when $t$ (and hence $h_{t}$) goes to infinity.

For all $i\geq 1$, the two processes $(\tilde{V}^{(i)}(\tilde{m}_{i}-x),\ 0\leq x\leq\tilde{m}_{i}-\tilde{\tau}_{i}^{-}(h_{t}))=\tilde{P}^{(i)}$ and $(\tilde{V}^{(i)}(\tilde{m}_{i}+x),\ 0\leq x\leq\tilde{\tau}_{i}(h_{t})-\tilde{m}_{i})$ are independent and the second is equal in law to $(V^{\uparrow}(x),\ 0\leq x\leq\tau(V^{\uparrow},h_{t}))$.

Let us now recall a fact from [46]:

Fact 4.8**.**

Assume that the hypotheses of Theorems 1.4 and 1.9 are satisfied. Fix $\epsilon$ small enough. There is a positive constant $c$ (depending on $\epsilon$) such that for all $t$ large enough

[TABLE]

where $A^{j}$ is defined in Subsection 2.1.

Proof.

(4.127) and (4.128) are respectively Lemmas 4.6 and 4.7 from [46]. For (4.129), note that

[TABLE]

Then, $|A^{j}(\tilde{\tau}_{j}^{-}(h_{t}/2))|$ and $A^{j}(\tilde{\tau}_{j}^{+}(h_{t}/2))$ can be bounded as in the proof of Lemma 4.8 of [46] which yields (4.129).

∎

4.2. Contribution of the valleys to the traveled distance

We need to approximate the contribution to the distance traveled by the diffusion in each valley. We have:

Proposition 4.9**.**

On an enlarged probability space, there is an iid sequence $(D_{j}^{t})_{j\geq 1}$, independent from the sequence $(e_{j},S_{j}^{t},R_{j}^{t})_{j\geq 1}$ and such that for $t$ large enough,

[TABLE]

where $q$ is the constant in Theorem 1.4 of [46] and $c_{1},c_{2}$ are positive constants.

Proof.

From the definitions of valleys we have

[TABLE]

Recall the definition of $m^{*}(h_{t})$ from Section 3.5 in [46]:

[TABLE]

From the Markov property at the stopping times $\tilde{L}_{j-1}$, $\tilde{L}^{\sharp}_{j}$ and $\tilde{\tau}_{j}(h_{t})$, and Fact 4.7 we get that the terms in the right hand side of (4.130) are respectively equal in law to $\tau(V,]-\infty,-e^{(1-\delta)\kappa h_{t}}])$, $m^{*}(h_{t})$, $\tau(V^{\uparrow},h_{t})$ and $\tau(V,]-\infty,-h_{t}/2])$. For the first and fourth term, applying Lemma 4.1 with $y=e^{(1-\delta)\kappa h_{t}}$ and $r=e^{(1-\delta/2)\kappa h_{t}}/2$ we get for $t$ large enough,

[TABLE]

For the third term, applying (4.122) with $x=0$, $y=h_{t}$ and $r=e^{(1-\delta/2)\kappa h_{t}}$ we get for $t$ large enough,

[TABLE]

The most important term in (4.130) is the second one, its law is given by Lemma 3.9 of [46]. Before studying it, let us recall some of the notations used in Section 3.5 of [46].

Let $\mathcal{F}$ denote the space of excursions, that is, càd-làg functions from $[0,+\infty[$ to $\mathbb{R}$, starting at zero and killed at the first positive instant when they reach [math] (this instant can possibly be infinite). For $\xi\in\mathcal{F}$, let us denote $\zeta(\xi):=\inf\{s>0,\ \xi(s)=0\}$ for the length of the excursion $\xi$. For $h>0$, let $\mathcal{F}_{h,-}$ and $\mathcal{F}_{h,+}$ denote respectively the set of excursions whose height is strictly less than $h$ and the set of excursions higher than $h$:

[TABLE]

Let $\mathcal{N}$ be the measure defined on $\mathcal{F}$ as in Subsection 3.5 of [46]. Let $S^{h,-}$ and $S^{h,+}$ be two independent pure jumps subordinators with Lévy measures respectively $\zeta\mathcal{N}(\mathcal{F}_{h,-}\cap.)$ and $\zeta\mathcal{N}(\mathcal{F}_{h,+}\cap.)$, the image measures of respectively $\mathcal{N}(\mathcal{F}_{h,-}\cap.)$ and $\mathcal{N}(\mathcal{F}_{h,+}\cap.)$ by $\zeta$. The sum of these measures equals $\zeta\mathcal{N}$ so $S:=S^{h,-}+S^{h,+}$ is a pure jumps subordinator with Lévy measure $\zeta\mathcal{N}$. Let also $T_{h}$ be an exponential random variable with parameter $\mathcal{N}(\mathcal{F}_{h,+})$, independent from $S^{h,-}$ and $S^{h,+}$. According to Lemma 3.9 of [46] we have

[TABLE]

Recall that the sequence $(\tilde{m}_{j}-\tilde{L}^{\sharp}_{j})_{j\geq 1}$ is iid because of Remark 2.2. Therefore, on an enlarged probability space, there is an iid sequence $(S^{h_{t},-}_{j},T_{h_{t},j})_{j\geq 1}$ such that for all $j\geq 1$,

[TABLE]

Let $d:=\mathcal{N}(\mathcal{F}_{h_{t},+})T_{h_{t}}$ and more generally $d_{j}:=\mathcal{N}(\mathcal{F}_{h_{t},+})T_{h_{t},j}$. Then, $d_{j}\overset{\mathcal{L}}{\sim}\mathcal{E}(1)$ and the sequence $(d_{j})_{j\geq 1}$ is iid. For large $t$, it is natural to approximate $S^{h_{t},-}(T_{h_{t}})$ by a multiple of $d$, for this we write

[TABLE]

For the first term, using the expression of $S$ in terms of $S^{h_{t},+}$ and $S^{h_{t},-}$, the independence between $S^{h_{t},+}$ and $T_{h_{t}}$, the definition of $S^{h_{t},+}$, Cauchy-Schwarz’s inequality and the definition of $S$, we have

[TABLE]

$Var(S(1))$ is indeed finite thanks to Lemma 3.10 of [46]. For the second term in the right hand side of (4.134), using Cauchy-Schwarz’s inequality and the independence between $S$ and $T_{h_{t}}$,

[TABLE]

For the third term in the right hand side of (4.134), using the definition of $d$,

[TABLE]

Then, recall the expression of $q$ in Proposition 3.13 of [46] (see also the proof of that proposition for the expression of $q$ involving $\mathbb{E}[S(1)]$) and Lemma 3.11 of [46]:

[TABLE]

and

[TABLE]

We thus get

[TABLE]

Let us define $D^{t}:=e^{\kappa h_{t}}d/q$ and more generally $D_{j}^{t}:=e^{\kappa h_{t}}d_{j}/q$. We then have indeed $D_{j}^{t}\overset{\mathcal{L}}{\sim}\mathcal{E}(e^{-\kappa h_{t}}q)$ and the fact that the sequence sequence $(D_{j}^{t})_{j\geq 1}$ is iid. (4.138) becomes

[TABLE]

where $C$ is a positive constant. Using Markov’s inequality, (4.135) and (4.136), and the asymptotic of $\mathcal{N}(\mathcal{F}_{h_{t},+})$ given by (4.137), we get the existence of a positive constant $C$ such that for $t$ large enough,

[TABLE]

[TABLE]

As a consequence there is a positive constant $C$ such that for $h_{t}$ large enough,

[TABLE]

Combining with (4.139) and putting into (4.134) we get

[TABLE]

Combining with (4.133) we get

[TABLE]

Then, we have $\mathbb{P}(D^{t}_{j}\leq e^{(1-\delta/2)\kappa h_{t}})=\mathbb{P}(d_{j}\leq qe^{-\delta\kappa h_{t}/2})\leq qe^{-\delta\kappa h_{t}/2}$ so putting into (4.131) and (4.132) we get for all $j\geq 1$

[TABLE]

Combining (4.140) and (4.141), and putting into (4.130) we get the result for $t$ large enough.

∎

4.3. Proof of some facts and lemmas

This subsection is devoted to the justification of the facts stated in Subsection 2.1, which mainly come from [46]. As these results are included in Section 2, we prove them under the hypotheses of Theorems 1.4 and 1.9: $0<\kappa<1$, $V$ has unbounded variation and there exists $p>1$ such that $V(1)\in L^{p}$.

In the facts and lemmas considered here, the value of the constant $c$ is not important (as long as it is positive) so we allow it to decrease from line to line and it is implicit that the estimates are true for $t$ large enough.

Proof.

of Lemma 2.1

(2.24) is only Lemma 4.17 of [46]. For (2.25), note that $\{N_{t}\leq\tilde{n}_{t}\}=\{H(m_{\tilde{n}_{t}+1})>t\}$ and $H(\tilde{m}_{\tilde{n}_{t}+1})=H(m_{\tilde{n}_{t}+1})$ on $\{V\in\mathcal{V}_{t}\}$ (since $\tilde{n}_{t}<n_{t}$). Let $t$ be large enough so that $(1-2/\log(h_{t}))(1+e^{-\tilde{c}h_{t}})^{-1}\geq 1/2$. Using the definitions of $\mathcal{E}^{5}_{t}$ and $\mathcal{E}^{7}_{t}$ (which is possible here since $\tilde{n}_{t}<n_{t}$) we get that $\mathbb{P}(\{N_{t}\leq\tilde{n}_{t}\}\cap\{V\in\mathcal{V}_{t}\}\cap\mathcal{E}^{5}_{t}\cap\mathcal{E}^{7}_{t})$ is less than

[TABLE]

According to (2.40) applied with some $\eta\in]0,(1-\rho)/3[$ we have that

[TABLE]

where $\mathcal{C}^{\prime}$ is the constant in Fact 2.8. Since $\rho<1$ the later converges to [math] so

[TABLE]

Let $\rho^{\prime}:=-(\rho+\kappa(\rho-1))$. $\rho^{\prime}$ is positive thanks to the hypothesis $\rho\in]0,\kappa/(1+\kappa)[$ so $e^{(\rho+\kappa(\rho-1))\phi(t)}$ converges to [math] and we deduce that for $t$ large enough,

[TABLE]

Combining with the bounds for $\mathbb{P}(V\notin\mathcal{V}_{t})$ and $\mathbb{P}(\overline{\mathcal{E}^{5}_{t}})+\mathbb{P}(\overline{\mathcal{E}^{7}_{t}})$ given by respectively Fact 2.3 and (2.27) we get (2.25). ∎

Now, let us recall a fact about Brownian local time:

Fact 4.10**.**

There is a positive constant $c$ such that for all $t$ large enough

[TABLE]

Proof.

(4.142) comes from the second Ray-Knight Theorem combined with estimate $(7.15)$ of [5] applied with $u=e^{-h_{t}/20}$, $v=e^{-h_{t}/8}$. (4.143) comes from the second Ray-Knight Theorem combined with estimate $(7.16)$ of [5].

∎

Proof.

of Fact 2.5

According to the combination of Lemmas 5.21 and 5.22 of [46], there is a positive constant $c$ such that $\mathbb{P}(\overline{\mathcal{E}^{1}_{t}})\leq e^{-ch_{t}}$ for large $t$. Since $h_{t}$ is ultimately greater than $\phi(t)$ we get $\mathbb{P}(\overline{\mathcal{E}^{1}_{t}})\leq e^{-c\phi(t)}$ for large $t$.

$\mathbb{P}(\overline{\mathcal{E}^{2}_{t}})\leq e^{-c\phi(t)}$ and $\mathbb{P}(\overline{\mathcal{E}^{3}_{t}})\leq e^{-c\phi(t)}$, for $c$ possibly decreased and large $t$, come from Fact 4.4 of [46] (respectively the second and third point).

For $\mathbb{P}(\overline{\mathcal{E}^{4}_{t}})$, note that $\mathcal{L}_{X}(H(\tilde{L}_{j}),.)-\mathcal{L}_{X}(H(\tilde{m}_{j}),.)=\mathcal{L}_{X_{\tilde{m}_{j}}}(H_{X_{\tilde{m}_{j}}}(\tilde{L}_{j}),.)$. Since $\tilde{c}>0$ has been chosen "small enough" in the Introduction, we can assume $\tilde{c}\leq 1/9$ so $\mathbb{P}(\overline{\mathcal{E}^{4}_{t}})\leq e^{-ch_{t}}\leq e^{-c\phi(t)}$, for $c$ possibly decreased and large $t$, follows from Fact 4.12 of [46] (and the fact that $h_{t}$ is ultimately greater than $\phi(t)$).

$\mathbb{P}(\overline{\mathcal{E}^{5}_{t}})\leq e^{-c\phi(t)}$, for $c$ possibly decreased and large $t$, comes from Fact 4.3 of [46].

$\tilde{c}>0$ has been fixed "small enough" in the Introduction. We can assume that it was chosen so small such that Proposition 4.5 of [46] apply with $\tilde{c}$ instead the constant $\epsilon/7$ there. $\mathbb{P}(\overline{\mathcal{E}^{6}_{t}})\leq e^{-ch_{t}}\leq e^{-c\phi(t)}$ and $\mathbb{P}(\overline{\mathcal{E}^{7}_{t}})\leq e^{-ch_{t}}\leq e^{-c\phi(t)}$, for $c$ possibly decreased and large $t$, thus come from Proposition 4.5 of [46] (and the fact that $h_{t}$ is ultimately greater than $\phi(t)$).

The second point of the fact is Lemma 4.11 of [46].

∎

Proof.

of Lemma 2.6

According to (2.41) and Chernoff inequality we have for any $j\geq 1$

[TABLE]

We have

[TABLE]

where, for the last two inequalities, we used Lemma 2.13 (applied with $z_{t}=e^{h_{t}^{1/3}}$, $a=2$) and (1.8), and where $K_{0}$ is the constant in (1.8). The term $2e^{-\delta\kappa h_{t}/3}$ is only necessary in the case $V=W_{\kappa}$ (because of (2.52)). Combining (4.144) and (4.145) we get that (2.30) is satisfied with probability at least $1-e^{-ch_{t}}$.

According to Fact 4.8 we have that (2.31), (2.32), (2.33) and (2.34) are satisfied with probability at least $1-e^{-ch_{t}}$.

According to (2.27) we have $\mathbb{P}(\cup_{i=1}^{7}\overline{\mathcal{E}^{i}_{t}})\leq e^{-L\phi(t)}$. Then,

[TABLE]

We thus deduce that (2.35) is satisfied with probability at least $1-e^{-L\phi(t)/2}$. Combing all this with Fact 2.3, we get Lemma 2.6 for a suitably chosen constant $c$. ∎

Proof.

of Fact 2.7

The ideas are very similar to the ones used for the proof of Lemma 5.3 of [5]. Since we are in the more general context of a Lévy potential and since we do not prove exactly the same thing (here, we work conditionally to the environment), we give the details.

Recall that $\epsilon>0$ has been fixed in the definition of $\mathcal{G}_{t}$. $\tilde{c}>0$ has been fixed "small enough" in the Introduction (and we already fixed some constraints about how small it must be in the proof of Fact 2.5). We can assume further that it was chosen so small such that $\tilde{c}<\min(1/20,\epsilon/4)$. Let us fix an environment $v\in\mathcal{G}_{t}$, $z\in[0,1-4/\log(h_{t})]$ and $k\leq n_{t}$. We put $\tilde{x}_{t}:=(1+e^{-\tilde{c}h_{t}})^{-1}x_{t}$, $Z:=(1-z)/(1-e^{-\tilde{c}h_{t}})R_{k}^{t}$. To study $\mathcal{E}^{8}_{t}(v,k,z)$, we look at its intersections with the events $\{\sigma_{X_{\tilde{m}_{k}}}(t\tilde{x}_{t},\tilde{m}_{k})<H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})\}$ and $\{\sigma_{X_{\tilde{m}_{k}}}(t\tilde{x}_{t},\tilde{m}_{k})>H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})\}$. We have

[TABLE]

where $\mathcal{D}_{k}$ is defined in (2.26). On the above event, $\sigma_{X_{\tilde{m}_{k}}}(tZ,\tilde{m}_{k})$ is finite and the diffusion stays in $[\tilde{L}_{k-1},\tilde{L}_{k}]$ until this time. $\sigma_{X_{\tilde{m}_{k}}}(tZ,\tilde{m}_{k})$ is thus equal to

[TABLE]

where $B$ is the Brownian motion driving the diffusion $X_{\tilde{m}_{k}}$ and $A^{k}$ is defined in Subsection 2.1. Let $\tilde{B}:=B((tZ)^{2}.)/tZ$ and $\tilde{A}^{k}(.):=A^{k}(.)/tZ$. From the scaling property $\tilde{B}$ is still a Brownian motion. We can write

[TABLE]

Since $v\in\mathcal{G}_{t}$ it satisfies (2.30). Recall also that $t\geq e^{h_{t}}$. For $t$ large enough so that $4e^{-h_{t}/8}/(1-e^{-\tilde{c}h_{t}})\log(h_{t})\geq e^{-h_{t}/4}$ we thus have $tZ\geq e^{3h_{t}/4}$ for any choice of $z\in[0,1-4/\log(h_{t})]$. Combining this with the fact that $v$ satisfies (2.31) and (2.32) we get

[TABLE]

Combining with (4.142) we get

[TABLE]

and we deduce that

[TABLE]

Since $v\in\mathcal{G}_{t}$ it satisfies (2.33) and (2.34). Combining with (4.143) (applied with $u=e^{\epsilon h_{t}/2}$) and the lower bound for $R_{k}^{t}$ in (2.30) we get

[TABLE]

Now putting (4.150), (4.151) and (4.152) into (4.148) (and combining with the fact that $\tilde{c}<\min(1/20,\epsilon/4)$) we get

[TABLE]

Combining with the definition of $Z$ we see that $t(1-z)\leq I$ on the complementary of $\mathcal{E}_{neg}^{1}$. Combining this with (4.146) we get that $\mathcal{E}^{8}_{t}(v,k,z)\cap\{\sigma_{X_{\tilde{m}_{k}}}(t\tilde{x}_{t},\tilde{m}_{k})<H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})\}$ is included into

[TABLE]

On the main event, $\sigma_{X_{\tilde{m}_{k}}}(tZ,\tilde{m}_{k})$ is finite. On this event, $\mathcal{L}_{X_{\tilde{m}_{k}}}(\sigma_{X_{\tilde{m}_{k}}}(tZ,\tilde{m}_{k}),y)$ is thus equal to

[TABLE]

Here again, for $\tilde{B}:=B((tZ)^{2}.)/tZ$ and $\tilde{A}^{k}(.):=A^{k}(.)/tZ$. We have

[TABLE]

Since $\mathcal{D}_{k}\subset[\tilde{\tau}_{k}^{-}(h_{t}/2),\tilde{\tau}_{k}^{+}(h_{t}/2)]$ we can apply (4.149) (together with $\tilde{c}<1/20$) and get

[TABLE]

On the big event in (4.153) we thus have both $(1+e^{-\tilde{c}h_{t}})tZ>tx_{t}$ and $tZ\leq t\tilde{x}_{t}$, (expect possibly on the event $\mathcal{E}_{neg}^{2}$). Since these two inequalities are not compatible we get

[TABLE]

We now study the case where $\sigma_{X_{\tilde{m}_{k}}}(t\tilde{x}_{t},\tilde{m}_{k})>H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})$. First, we have

[TABLE]

Indeed, $\sup_{\mathcal{D}_{k}}\mathcal{L}_{X_{\tilde{m}_{k}}}(t(1-z),.)\leq\sup_{\mathcal{D}_{k}}\mathcal{L}_{X_{\tilde{m}_{k}}}(H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k}),.)$ on $\{t(1-z)\leq H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})\}$ and $\sup_{\mathcal{D}_{k}}\mathcal{L}_{X_{\tilde{m}_{k}}}(t(1-z),.)=\sup_{\mathcal{D}_{k}}\mathcal{L}_{X_{\tilde{m}_{k}}}(H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k}),.)$ on $\{H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})<t(1-z)\}\cap\mathcal{E}^{1}_{t}$. Since we are dealing with $X_{\tilde{m}_{k}}$, the diffusion shifted at time $H(\tilde{m}_{k})$, we can see that the main event above is included in $\overline{\mathcal{E}^{4}_{t}}$. We thus get

[TABLE]

(2.37) follows easily from the combination of (4.155) and (4.156) applied with $z=1-4/\log(h_{t})$. Then, the right hand sides of (4.155) and (4.156) do not depend on $z$ (which is arbitrary in $[0,1-4/\log(h_{t})]$) and $H(\tilde{m}_{k})/t$ is independent from $X_{\tilde{m}_{k}}$. We can thus replace $z$ by $H(\tilde{m}_{k})/t$ in (4.155) and (4.156) (at least on $\{N_{t}\geq k,\ H(\tilde{m}_{k})/t\leq 1-4/\log(h_{t})\}$). Using the combination of (4.155) and (4.156) to study the union of events in (2.36) we get

[TABLE]

Since $v\in\mathcal{G}_{t}$ it satisfies (2.35). We thus have $P^{v}(\overline{\mathcal{E}^{1}_{t}}\cup\overline{\mathcal{E}^{4}_{t}})\leq e^{-L\phi(t)/2}$ (where $L$ is the constant defined in Fact 2.5) and from the definition of $h_{t}$ and $\phi(t)$ we have easily $2n_{t}e^{-ch_{t}}\leq e^{-c\phi(t)}$ for large $t$. For $t$ large enough we thus get (2.36).

We now prove (2.38). Let us fix an environment $v\in\mathcal{G}_{t}$, $z\in[0,1]$ and $k\leq n_{t}$. Inversing the local time in the definition of $\mathcal{E}^{9}_{t}(v,k,z)$ we get that $\mathcal{E}^{9}_{t}(v,k,z)$ coincides with the event

[TABLE]

We have to distinguish the cases $H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})>\sigma_{X_{\tilde{m}_{k}}}(t/x_{t},\tilde{m}_{k})$ and $\sigma_{X_{\tilde{m}_{k}}}(t/x_{t},\tilde{m}_{k})>H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})$. On $\mathcal{E}^{9}_{t}(v,k,z)\cap\{\sigma_{X_{\tilde{m}_{k}}}(t\tilde{x}_{t},\tilde{m}_{k})<H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})\}$, $\sigma_{X_{\tilde{m}_{k}}}(t/x_{t},\tilde{m}_{k})$ is finite and the diffusion stays in $[\tilde{L}_{k-1},\tilde{L}_{k}]$ until this time. On this event, $\sigma_{X_{\tilde{m}_{k}}}(t/x_{t},\tilde{m}_{k})$ is thus equal to

[TABLE]

where, as in (4.147), $B$ is the Brownian motion driving the diffusion $X_{\tilde{m}_{k}}$. Now, let $\tilde{B}:=B((t/x_{t})^{2}.)/(t/x_{t})$ and $\tilde{A}^{k}(.):=A^{k}(.)/(t/x_{t})$. From the scaling property $\tilde{B}$ is still a Brownian motion. We can write

[TABLE]

From $t\geq e^{h_{t}}$, the definition (2.29) of $x_{t}$ and the definition (2.23) of $h_{t}$ we see that $t/x_{t}\geq e^{3h_{t}/4}$ for $t$ large enough. Recall also that $v\in\mathcal{G}_{t}$ implies that $v$ satisfies (2.31) and (2.32). We thus get

[TABLE]

We can now proceed as in the proof of (2.36) to get analogues of (4.150), (4.151) and (4.152). We deduce that

[TABLE]

Now, note from (4.157) that $\mathcal{E}^{9}_{t}(v,k,z)\subset\{(1-e^{-\tilde{c}h_{t}})\sigma_{X_{\tilde{m}_{k}}}(t/x_{t},\tilde{m}_{k})\geq tR_{k}^{t}/x_{t}\}$. Since

$\sigma_{X_{\tilde{m}_{k}}}(t/x_{t},\tilde{m}_{k})=I^{\prime}$ on $\mathcal{E}^{9}_{t}(v,k,z)\cap\{\sigma_{X_{\tilde{m}_{k}}}(t/x_{t},\tilde{m}_{k})<H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})\}$ and $(1+e^{-\tilde{c}h_{t}})(1-e^{-\tilde{c}h_{t}})<1$ we deduce that $\mathcal{E}^{9}_{t}(v,k,z)\cap\{\sigma_{X_{\tilde{m}_{k}}}(t/x_{t},\tilde{m}_{k})<H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})\}\subset\mathcal{E}_{neg}^{3}$. Then,

[TABLE]

We now study the case where $\sigma_{X_{\tilde{m}_{k}}}(t/x_{t},\tilde{m}_{k})>H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})$, following the ideas of the proof of Lemma 5.3 of [5]. Recall that $B$ is the Brownian motion driving the diffusion $X_{\tilde{m}_{k}}$. We have

[TABLE]

Now, let $\tilde{B}:=B((A^{k}(\tilde{L}_{k}))^{2}.)/A^{k}(\tilde{L}_{k})$. From the scaling property $\tilde{B}$ is still a Brownian motion. $\mathcal{L}_{B}\left(\tau(B,A^{k}(\tilde{L}_{k})),0\right)=A^{k}(\tilde{L}_{k})\mathcal{L}_{\tilde{B}}\left(\tau(\tilde{B},1),0\right)$. Note that from the definition of $e_{k}$ given in Subsection 2.1 we have $\mathcal{L}_{\tilde{B}}\left(\tau{\tilde{B}}(1),0\right)=e_{k}$. As a consequence,

[TABLE]

Using the trivial inequality

[TABLE]

and the definition of $\mathcal{E}^{7}_{t}$ we deduce that

[TABLE]

Now, note from (4.157) that $\mathcal{E}^{9}_{t}(v,k,z)\subset\{(1-e^{-\tilde{c}h_{t}})H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})\geq tR_{k}^{t}/x_{t}\}$. Since $(1+e^{-\tilde{c}h_{t}})(1-e^{-\tilde{c}h_{t}})<1$, the inequality $tR_{k}^{t}/x_{t}>(1+e^{-\tilde{c}h_{t}})^{-1}H_{X_{\tilde{m}_{k}}}(\tilde{L}_{k})$ is in contradiction with the event $\mathcal{E}^{9}_{t}(v,k,z)$. We thus get

[TABLE]

Since the right hand sides of (4.158) and (4.159) do not depend on $z$ (which is arbitrary in $[0,1]$) and $H(\tilde{m}_{k})/t$ is independent from $X_{\tilde{m}_{k}}$, we can replace $z$ by $H(\tilde{m}_{k})/t$ in (4.158) and (4.159) (at least on $\{N_{t}\geq k\}$). We can thus use the combination of (4.158) and (4.159) to study the union of events in (2.38). We get

[TABLE]

Since $v\in\mathcal{G}_{t}$ it satisfies (2.35). We thus have $P^{v}(\overline{\mathcal{E}^{7}_{t}})\leq e^{-L\phi(t)/2}$ (where $L$ is the constant defined in Fact 2.5) and from the definition of $h_{t}$ and $\phi(t)$ we have easily $n_{t}e^{-ch_{t}}\leq e^{-c\phi(t)}$ for large $t$. For $t$ large enough we thus get (2.38).

∎

Proof.

of Fact 2.8

(2.39) and (2.40) are included into Lemma 4.16 of [46] while (2.41) comes from Proposition 3.8 of [46] applied with $h=h_{t}$. ∎

Proof.

of Lemma 2.9

First, note that from (2.23) and the definition of $n_{t}$ just after, we have $2t^{\kappa}e^{\kappa\delta(\log(\log(t)))^{\omega}}/q=2n_{t}e^{\kappa h_{t}}/q$. From the definition of $N_{t}$, we know that on $\{V\in\mathcal{V}_{t}\}$, at time $t$ , $\tilde{m}_{N_{t}+1}$ has never been reached by the diffusion and neither $\tilde{L}_{N_{t}+1}$ (because $\tilde{L}_{N_{t}+1}>\tilde{m}_{N_{t}+1}$), we thus have

[TABLE]

where $c$ is a positive constant, $c_{1}$ has the same meaning as in Proposition 4.9 and $t$ is large enough. For the last inequality, we used Proposition 4.9, Fact 2.3 and (2.24).

[TABLE]

where we used Markov’s inequality, the fact that the sequence $(qe^{-\kappa h_{t}}D_{j}^{t})$ is an iid sequence of exponential random variable with parameter $1$ and the expression of the Laplace transform for the exponential distribution. Since $\log(2)<1$ and $n_{t}=e^{\kappa(1+\delta)(\log(\log(t)))^{\omega}}>>h_{t}$, we get the first point for $t$ large enough.

We now prove the second point. We first note that from (2.23) and the definition of $\tilde{n}_{t}$ just after, we have $t^{\kappa}e^{(\rho-\kappa)(\log(\log(t)))^{\omega}}/2q\underset{t\rightarrow+\infty}{\sim}\tilde{n}_{t}e^{\kappa h_{t}}/2q$. On $\{V\in\mathcal{V}_{t}\}\cap\{N_{t}<n_{t}\}\cap\mathcal{E}^{1}_{t}$ we have $X(t)\geq\tilde{L}_{N_{t}-1}$ so

[TABLE]

where $c$ is a positive constant, $c_{1}$ has the same meaning as in Proposition 4.9 and $t$ is large enough. For the last inequality we used Proposition 4.9, Fact 2.3, Lemma 2.1, (2.27) and the fact that $e^{-ch_{t}}\leq e^{-c\phi(t)}$ for large $t$.

[TABLE]

where we used Markov’s inequality, the fact that the sequence $(qe^{-\kappa h_{t}}D_{j}^{t})_{j\geq 1}$ is an iid sequence of exponential random variable with parameter $1$ and the expression for the Laplace transform of the exponential distribution. Since $\log(3)>1$ and $\tilde{n}_{t}=\lfloor e^{\rho(\log(\log(t)))^{\omega}}\rfloor>>h_{t}$, we get the second point for $t$ large enough.

We now prove (2.44). Note that for a fixed environment $v\in\mathcal{V}$ we have

[TABLE]

and note that

[TABLE]

where the equality in law comes from the time-reversal property. Applying Lemma 4.2 with $t=r/2$ and $a=\sqrt{r}$ we get

[TABLE]

where $c_{1}$ and $c_{2}$ are the constants in the lemma. Then, applying Lemma 4.3 we get for $r$ large enough,

[TABLE]

Putting (4.161) and (4.162) into (4.160) we get that with $P$-probability greater than $1-(e^{c_{1}\sqrt{r}-c_{2}r/2}+e^{-\kappa\sqrt{r}}+e^{-\kappa\sqrt{r}/2})$: $P^{V}\left(\inf_{[0,+\infty[}X\leq-r\right)\leq 2r^{-1}$ (and it is bounded by $1$ when this estimates fails) so integrating on $\mathcal{V}$ with respect to $P$ we get

[TABLE]

(2.44) follows for $r$ large enough. Finally, (2.45) is included in Lemma 5.19 of [46].

∎

4.4. Almost sure constantness of $\limsup$ and $\liminf$

We now use a classical argument involving Kolmogorov $0-1$ law to justify the almost sure constantness of the $\limsup$ stated in Remark 1.3. We here treat the case of the $\limsup$ with the renormalization $t\log(\log(t))$. The same argument can be used with the $\liminf$ instead of the $\limsup$, or with any of the other renormalizations used in the paper.

We first fix $v\in\mathcal{V}$, a realization of the environment. For any $n\in\mathbb{N}$, the process $X^{n}:=X(\tau(X,n)+.)$ is, according to the Markov property, a diffusion in the environment $v(n+.)$ and it is independent from $(X(s),\ 0\leq s\leq\tau(X,n))$. We have

[TABLE]

where the first equality comes from the fact that $\mathbb{P}$-almost surely the favorite site $F^{*}(t)$ goes to $+\infty$. Indeed, the diffusion $\mathbb{P}$-almost surely converges to $+\infty$ and as we can see from the results of Subsection 1.1, $\mathcal{L}_{X}^{*}(t)$ converges $\mathbb{P}$-almost surely to infinity, these two facts imply the convergence of $F^{*}(t)$ to $+\infty$. As a consequence $F^{*}(t)$ will become greater than $n$ for $t$ large enough which implies the first equality in (4.163). The second equality comes from the equivalence when $t$ goes to infinity between $(\tau(X,n)+t)\log(\log(\tau(X,n)+t))$ and $t\log(\log(t))$.

The $\limsup$ in the right hand side of (4.163) belongs to the $\sigma$-field $\sigma(X^{n}(t),\ t\geq 0$, it is thus independent from $\sigma(X(s),\ 0\leq s\leq\tau(X,n))$. Since this is true for any $n\in\mathbb{N}$ we get, according to Kolmogorov $0-1$ law, that the $\limsup$ is constant $P^{v}$-almost surely. In other words, the $\limsup$ is only a deterministic function of the environment $v$, let us denote it by $L(v)$.

Let us fix $v\in\mathcal{V}$ and $n\in\mathbb{N}$. Note that $P^{v}(\inf X^{n}\geq n-1)>0$ so the $\limsup$ still equals $L(v)$ on $\{\inf X^{n}\geq n-1\}$. As a consequence, $L(v)$ is only a function of $(v(x+n-1)-v(n-1),\ x\geq 0)$. If we consider the space $\mathcal{V}$ equipped with probability $P$, this implies that $L(V)$ is independent from the $\sigma$-field $\sigma(V(x),\ x\leq n-1)$. Since this is true for all $n\in\mathbb{N}$ we deduce from Kolmogorov $0-1$ law that $L(V)$ is constant $P$-almost surely. This proves that the $\limsup$ is in fact constant $\mathbb{P}$-almost surely. In other words,

[TABLE]