On penalisation results related with a remarkable class of submartingales

TL;DR

This paper extends penalization results for certain diffusions, showing convergence of probability measures and identifying a universal sigma-finite measure related to the original diffusion measure.

Contribution

It generalizes previous results by establishing convergence of penalized diffusions to a limit measure absolutely continuous with respect to a universal sigma-finite measure.

Findings

Convergence of penalized measures to a limit measure

Existence of a universal sigma-finite measure

General framework for measure relations

Abstract

Is this paper we study penalisations of diffusions satisfying some technical conditions, generalizing a result obtained by Najnudel, Roynette and Yor. If one of these diffusions has probability distribution $\mathbb{P}$, then our result can be described as follows: for a large class of families of probability measures $(\mathbb{Q}_t)_{t \geq 0}$, each of them being absolutely continuous with respect to $\mathbb{P}$, there exists a probability $\mathbb{Q}_{\infty}$ such that for all events $\Lambda$ depending only on the canonical trajectory up to a fixed time, $\mathbb{Q}_t (\Lambda)$ tends to $\mathbb{Q}_{\infty} (\Lambda)$ when $t$ goes to infinity. In the cases we study here, the limit measure $\mathbb{Q}_{\infty}$ is absolutely continous with respect to a sigma-finite measure $\mathcal{Q}$, which does not depend on the choice of the family of probabilities $(\mathbb{Q}_t)_{t \geq 0}$, but only on $\mathbb{P}$. The relation between $\mathbb{P}$ and $\mathcal{Q}$ is obtained in a very general framework by the authors of this paper.

Equations

$$\mathcal{F}:=\underset{s\ge 0}{\bigvee }{\mathcal{F}}_s.$$

$$0<\mathbb{W}[{\Gamma }_t]<\infty ,$$

$${\mathbb{Q}}_t:=\frac{{\Gamma }_t}{\mathbb{W}[{\Gamma }_t]}.\mathbb{W}.$$

$${\mathbb{Q}}_t[{\Lambda }_s]\underset{t\to \infty }{⟶}{\mathbb{Q}}_{\infty }[{\Lambda }_s].$$

$$0<{\int }_{\mathbb{R}}(1+|x|)q(x)dx<\infty .$$

$${\Gamma }_t=\exp \left(-{\int }_{\mathbb{R}}(L_t^y{)}^{2}dy\right)$$

$${\mathbb{Q}}_t({\Lambda }_s)\underset{t\to \infty }{⟶}\mathbb{W}(M_s\text{mathds}{1}_{{\Lambda }_s}).$$

$${\mathbb{Q}}_t({\Lambda }_s)\underset{t\to \infty }{⟶}{\mathbb{Q}}_{\infty }^{(s)}({\Lambda }_s).$$

$$\bigcap _{n=0}^N{A}_n({\omega }_n)\ne \varnothing ,$$

$$\bigcap _{n=0}^{\infty }{A}_n({\omega }_n)\ne \varnothing .$$

$$\mathcal{Q}\left[F_s\text{mathds}{1}_{g\le s}\right]=\mathbb{P}\left[F_s{X}_s\right].$$

$$0<\mathbb{P}[{\Gamma }_t]<\infty$$

$$0<\mathcal{Q}[{\Gamma }_{\infty }]<\infty ,$$

$${\mathbb{Q}}_t:=\frac{{\Gamma }_t}{\mathbb{P}[{\Gamma }_t]}.\mathbb{P},$$

$${\mathbb{Q}}_{\infty }:=\frac{{\Gamma }_{\infty }}{\mathcal{Q}[{\Gamma }_{\infty }]}.\mathcal{Q}.$$

$$\mathbb{P}[X_{(T^{(0)}\wedge u)+s}|{\mathcal{F}}_{T^{(0)}\wedge u}]=X_{T^{(0)}\wedge u},$$

$$\mathbb{P}[X_{T^{(0)}+s}\text{mathds}{1}_{T^{(0)}\le u}]=\mathbb{P}[X_{T^{(0)}}\text{mathds}{1}_{T^{(0)}\le u}]=0,$$

$$g^{[a]}:=\sup \{s\ge 0,X_s\le a\},$$

$$0<\mathbb{P}[{\Gamma }_t]<\infty ,$$

$${\mathbb{Q}}_t:=\frac{{\Gamma }_t}{\mathbb{P}[{\Gamma }_t]}.\mathbb{P}.$$

$${\mathbb{Q}}_{\infty }:=\frac{{\Gamma }_{\infty }}{\mathcal{Q}[{\Gamma }_{\infty }]}.\mathcal{Q}$$

$${\mathbb{Q}}_t[{\Lambda }_s]\underset{t\to \infty }{⟶}{\mathbb{Q}}_{\infty }[{\Lambda }_s].$$

$${\Phi }_A^{\prime \prime }(t)=A(t){\Phi }_A(t)$$

$${\Theta }_{t_0,a,b}=a{\Theta }_{t_0,1,0}+b{\Theta }_{t_0,0,1}.$$

$$\Theta (t)={\Theta }_{1,1}(t)$$

$$\Theta (t)=\frac{1}{{\Theta }_{t,1}(1)}$$

$$\Theta (t)\le \Theta (1)+{\int }_t^1|{\Theta }^{\prime }(s)|ds\le 1+|{\Theta }^{\prime }(t)|\le L|{\Theta }^{\prime }(t)|$$

$${\Theta }^{\prime \prime }(t)\le LA(t)|{\Theta }^{\prime }(t)|,$$

$$\left|\frac{d}{dt}\log |{\Theta }^{\prime }(t)|\right|\le LA(t).$$

$${\Phi }_A(t)=\frac{\Theta (t)}{\Theta (0)}.$$

Full text

1. Introduction

In a series of articles by Roynette, Vallois and Yor, summarized in [15], the authors study many examples of probability measures on functional spaces, which are obtained as weak limits of measures which are absolutely continuous with respect to a given probability. The setting generally used is the following: one considers $\mathbb{W}$, the Wiener measure on the space of continuous functions from $\mathbb{R}_{+}$ to $\mathbb{R}$, denoted $\mathcal{C}(\mathbb{R}_{+},\mathbb{R})$ and endowed with its canonical filtration $(\mathcal{F}_{s})_{s\geq 0}$ (not completed). One defines the $\sigma$-algebra $\mathcal{F}$ by

[TABLE]

One then considers $(\Gamma_{t})_{t\geq 0}$, a family of nonnegative random variables on the same space such that

[TABLE]

and for $t\geq 0$, one defines the probability measure

[TABLE]

(In this paper, if $\mathbb{P}$ is a probability measure and $Y$ a random variable, we denote by $\mathbb{P}[Y]$ the expectation of $Y$ with respect to $\mathbb{P}$). Under these assumptions, Roynette, Vallois and Yor have proved that for many examples of families of functionals $(\Gamma_{t})_{t\geq 0}$, there exists a probability measure $\mathbb{Q}_{\infty}$ which can be considered as the weak limit of $(\mathbb{Q}_{t})_{t\geq 0}$ when $t$ goes to infinity, in the following sense: for all $s\geq 0$ and for all events $\Lambda_{s}\in\mathcal{F}_{s}$,

[TABLE]

For example, the measure $\mathbb{Q}_{\infty}$ exists for the following families of functionals $(\Gamma_{t})_{t\geq 0}$:

• •

$\Gamma_{t}=\phi(L_{t})$, where $(L_{t})_{t\geq 0}$ is the local time at zero of the canonical process $X$, and $\phi$ is a nonnegative, integrable function from $\mathbb{R}_{+}$ to $\mathbb{R}_{+}$.

• •

$\Gamma_{t}=\phi(S_{t})$, where $S_{t}$ is the supremum of $X$ on the interval $[0,t]$, and $\phi$ is, again, a nonnegative, integrable function from $\mathbb{R}_{+}$ to $\mathbb{R}_{+}$.

• •

$\Gamma_{t}=e^{-\int_{0}^{t}q(X_{s})ds}$, where $q$ is a measurable function from $\mathbb{R}$ to $\mathbb{R}_{+}$, such that:

[TABLE]

• •

$\Gamma_{t}=e^{\lambda L_{t}+\mu|X_{t}|}$, where $(L_{t})_{t\geq 0}$ is, again, the local time at zero of $X$.

There are still other interesting particular cases which can be studied. For instance in [8] Najnudel has proved that the limit measure exists for

[TABLE]

where $(L_{t}^{y})_{t\geq 0,y\in\mathbb{R}}$ is the regular family of local times of $X$. This example can be interpreted as the construction of a one-dimensional self-avoiding Brownian motion, and hence, a one-dimensional version of Edwards’ model, for polymers of infinite length (this model was studied with several points of view: see for example [6], [19], [3], [18]). Another family of penalisation of probability measures on general functional spaces has been introduced by the authors of this paper in [10], where the functional $(\Gamma_{t})$ is of the form $\Gamma_{t}=F_{t}X_{t}$, for a large class of functionals $(F_{t})$. Note that in all these examples, the proof of the existence of $\mathbb{Q}_{\infty}$ given by the authors cited above contains two steps which need to be clarified here. The first point is that the functional $\Gamma_{t}$ is, in general, not defined everywhere but only almost everywhere. For example, since $(\mathcal{F}_{s})_{s\geq 0}$ is not completed, there does not exist a càdlàg and adapted version of the local time which is defined everywhere (a more detailed discussion of this problem is given, for example, in our article [9]). However, the almost sure existence of $\Gamma_{t}$ is sufficient to define the measure $\mathbb{Q}_{t}$, and to study its weak convergence towards $\mathbb{Q}_{\infty}$. The second point is that the existence of $\mathbb{Q}_{\infty}$ depends on the possibility one has to extend compatible families of probability measures. More precisely, the penalisation results cited above are proved as follows: by studying the asymptotics (for fixed $s\geq 0$ and for $t$ going to infinity) of a suitable version of the conditional expectation of $\Gamma_{t}$ given $\mathcal{F}_{s}$, one proves that for all $s\geq 0$, there exists an $\mathcal{F}_{s}$-measurable, nonnegative random variable $M_{s}$ such that for all events $\Lambda_{s}\in\mathcal{F}_{s}$:

[TABLE]

One immediately deduces that $(M_{s})_{s\geq 0}$ is a martingale, and that there exists a compatible family $(\mathbb{Q}_{\infty}^{(s)})_{s\geq 0}$ of probability measures, $\mathbb{Q}_{\infty}^{(s)}$ defined on $\mathcal{F}_{s}$, such that for $s\geq 0$, $\Lambda_{s}\in\mathcal{F}_{s}$:

[TABLE]

Hence, if one can construct a probability measure on the space $(\mathcal{C}(\mathbb{R}_{+},\mathbb{R}),\mathcal{F})$, such that its restriction to $\mathcal{F}_{s}$ is $\mathbb{Q}_{\infty}^{(s)}$, the existence of $\mathbb{Q}_{\infty}$ is proved. This possibility of extension of measures is not obvious at all: for example, as explained in [9], $\mathbb{Q}_{\infty}$ does not exist in general if one replaces the filtered probability space $(\mathcal{C}(\mathbb{R}_{+},\mathbb{R}),\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{W})$ by its usual augmentation. However, in the setting described above, $\mathbb{Q}_{\infty}$ exists, because the filtered measurable space $(\mathcal{C}(\mathbb{R}_{+},\mathbb{R}),\mathcal{F},(\mathcal{F}_{s})_{s\geq 0})$ satisfies the property (P), described as follows:

Definition 1.1**.**

Let $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0})$ be a filtered measurable space, such that $\mathcal{F}$ is the $\sigma$-algebra generated by $\mathcal{F}_{s}$, $s\geq 0$: $\mathcal{F}=\bigvee_{s\geq 0}\mathcal{F}_{s}$. We say that the property (P) holds if and only if $(\mathcal{F}_{s})_{s\geq 0}$ enjoys the following conditions:

• •

For all $s\geq 0$, $\mathcal{F}_{s}$ is generated by a countable number of sets.

• •

For all $s\geq 0$, there exists a Polish space $\Omega_{s}$, and a surjective map $\pi_{s}$ from $\Omega$ to $\Omega_{s}$, such that $\mathcal{F}_{s}$ is the $\sigma$-algebra of the inverse images, by $\pi_{s}$, of Borel sets in $\Omega_{s}$, and such that for all $B\in\mathcal{F}_{s}$, $\omega\in\Omega$, $\pi_{s}(\omega)\in\pi_{s}(B)$ implies $\omega\in B$.

• •

If $(\omega_{n})_{n\geq 0}$ is a sequence of elements of $\Omega$, such that for all $N\geq 0$,

[TABLE]

where $A_{n}(\omega_{n})$ is the intersection of the sets in $\mathcal{F}_{n}$ containing $\omega_{n}$, then:

[TABLE]

This definition is given in [9], but the corresponding conditions are not new: they are already stated by Parthasarathy in [14], p. 141. A fundamental example of filtered measurable space $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0})$ satisfying the property (P) is the following: for some integer $d\geq 1$, $\Omega$ is the space of continuous functions from $\mathbb{R}_{+}$ to $\mathbb{R}^{d}$, or the space of càdlàg functions from $\mathbb{R}_{+}$ to $\mathbb{R}^{d}$, for all $s\geq 0$, $\mathcal{F}_{s}$ is the $\sigma$-algebra generated by the canonical process up to time $s$, and $\mathcal{F}$ is the $\sigma$-algebra generated by $(\mathcal{F}_{s})_{s\geq 0}$. This example proves that in the examples studied by Najnudel, Roynette, Vallois and Yor, the measure $\mathbb{Q}_{\infty}$ can be constructed. Indeed, one has the following result, proved in [9], with methods coming from Stroock and Varadhan (see [17]):

Proposition 1.2**.**

Let $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0})$ be a filtered measurable space satisfying the property (P), and let, for $s\geq 0$, $\mathbb{Q}_{s}$ be a probability measure on $(\Omega,\mathcal{F}_{s})$, such that for all $t\geq s\geq 0$, $\mathbb{Q}_{s}$ is the restriction of $\mathbb{Q}_{t}$ to $\mathcal{F}_{s}$. Then, there exists a unique measure $\mathbb{Q}_{\infty}$ on $(\Omega,\mathcal{F})$ such that for all $s\geq 0$, its restriction to $\mathcal{F}_{s}$ is equal to $\mathbb{Q}_{s}$.

All the examples of penalisation results described above involve the Wiener space. In this paper, we need to deal with a more general setting, for which it is important to have, at the same time, the classical results of stochastic calculus generally proved under usual conditions, and the possibility of extending compatible families of probability measures. Since this extension is in general impossible under usual conditions, we need to complete more carefully the probability spaces. The good way to do this completion, intermediate between the right-continuous version and the usual augmentation, involves the so-called natural conditions or N-usual conditions. They were first introduced by Bichteler in [2], and then rediscovered in [9] where it is shown that most of the properties which are generally proved under usual conditions remain true under natural conditions (for example, existence of càdlàg versions of martingales, Doob-Meyer decomposition, début theorem, etc.). Let us recall here the definition:

Definition 1.3**.**

A filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$, satisfies the natural conditions iff the two following assumptions hold:

• •

The filtration $(\mathcal{F}_{s})_{s\geq 0}$ is right-continuous;

• •

For all $s\geq 0$, and for every $\mathbb{P}$-negligible set $A\in\mathcal{F}_{s}$, all the subsets of $A$ are contained in $\mathcal{F}_{0}$.

This definition is slightly different from the definitions given in [2] and [9] but one can easily check that it is equivalent. The natural enlargement of a filtered probability space can be defined by using the following proposition:

Proposition 1.4** ([9]).**

Let $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$ be a filtered probability space. There exists a unique filtered probability space $(\Omega,\widetilde{\mathcal{F}},(\widetilde{\mathcal{F}}_{s})_{s\geq 0},\widetilde{\mathbb{P}})$ (with the same set $\Omega$), such that:

• •

For all $s\geq 0$, $\widetilde{\mathcal{F}}_{s}$ contains $\mathcal{F}_{s}$, $\widetilde{\mathcal{F}}$ contains $\mathcal{F}$ and $\widetilde{\mathbb{P}}$ is an extension of $\mathbb{P}$;

• •

The space $(\Omega,\widetilde{\mathcal{F}},(\widetilde{\mathcal{F}}_{s})_{s\geq 0},\widetilde{\mathbb{P}})$ satisfies the natural conditions;

• •

For any filtered probability space $(\Omega,\mathcal{F}^{\prime},(\mathcal{F}^{\prime}_{s})_{s\geq 0},\mathbb{P}^{\prime})$ satisfying the two items above, $\mathcal{F}^{\prime}_{s}$ contains $\widetilde{\mathcal{F}}_{s}$ for all $s\geq 0$, $\mathcal{F}^{\prime}$ contains $\widetilde{\mathcal{F}}$ and $\mathbb{P}^{\prime}$ is an extension of $\widetilde{\mathbb{P}}$.

The space $(\Omega,\widetilde{\mathcal{F}},(\widetilde{\mathcal{F}}_{s})_{s\geq 0},\widetilde{\mathbb{P}})$ is called the natural enlargement of $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$.

Intuitively, the natural enlargement of a filtered probability space is its smallest extension which satisfies the natural conditions. Now, if we combine the natural enlargement with the property (P), we obtain the following definition:

Definition 1.5**.**

Let $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$ be a filtered probability space. We say that it satisfies the property (NP) iff it is the natural enlargement of a filtered probability space $(\Omega,\mathcal{F}^{0},(\mathcal{F}^{0}_{s})_{s\geq 0},\mathbb{P}^{0})$ such that the filtered measurable space $(\Omega,\mathcal{F}^{0},(\mathcal{F}^{0}_{s})_{s\geq 0})$ enjoys property (P).

The following result about extension of probability measures is proved in [9] (in a slightly more general form):

Proposition 1.6** ([9]).**

Let $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$ be a filtered probability space, satisfying property (NP). Then, the $\sigma$-algebra $\mathcal{F}$ is the $\sigma$-algebra generated by $(\mathcal{F}_{s})_{s\geq 0}$, and for all coherent families of probability measures $(\mathbb{Q}_{s})_{s\geq 0}$, such that $\mathbb{Q}_{s}$ is defined on $\mathcal{F}_{s}$, and is absolutely continuous with respect to the restriction of $\mathbb{P}$ to $\mathcal{F}_{s}$, there exists a unique probability measure $\mathbb{Q}$ on $\mathcal{F}$ which coincides with $\mathbb{Q}_{s}$ on $\mathcal{F}_{s}$ for all $s\geq 0$.

The possibility of extension of coherent families of probability measures, on spaces satisfying the property (NP) implies that one can obtain penalisation results in this framework. For the functionals we shall study in this article, the limit measure is absolutely continuous with respect to a remarkable $\sigma$-finite measure constructed in a very general setting in [11]. This $\sigma$-finite measure was already encountered in different special cases (see [1], [5], [7] and [12] for example), and it involves a remarkable class of submartingales, called $(\Sigma)$. The submartingales of class $(\Sigma)$ were first introduced by Yor in [20], and some of their main properties were studied in [13]. Let us recall the definition:

Definition 1.7**.**

Let $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$ be a filtered probability space. A nonnegative submartingale (resp. local submartingale) $(X_{s})_{s\geq 0}$ is of class $(\Sigma)$, if and only if it can be decomposed as $X_{s}=N_{s}+A_{s}$ where $(N_{s})_{s\geq 0}$ and $(A_{s})_{s\geq 0}$ are $(\mathcal{F}_{s})_{s\geq 0}$-adapted processes satisfying the following assumptions:

• •

$(N_{s})_{s\geq 0}$ is a càdlàg martingale (resp. local martingale);

• •

$(A_{s})_{s\geq 0}$ is a continuous increasing process, with $A_{0}=0$;

• •

The measure $(dA_{s})$ is carried by the set $\{s\geq 0,X_{s}=0\}$.

In [11], it is proved that for any submartingale of class $(\Sigma)$ defined on a space satisfying property (NP), on can associate with it a $\sigma$-finite measure as follows:

Theorem 1.8**.**

Let $(X_{s})_{s\geq 0}$ be a submartingale of the class $(\Sigma)$ (in particular $X_{s}$ is integrable for all $s\geq 0$), defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$ which satisfies the property (NP). In particular $(\mathcal{F}_{s})_{s\geq 0}$ satisfies the natural conditions and $\mathcal{F}$ is the $\sigma$-algebra generated by $\mathcal{F}_{s}$, $s\geq 0$. Then, there exists a unique $\sigma$-finite measure $\mathcal{Q}$, defined on $(\Omega,\mathcal{F},\mathbb{P})$, such that for $g:=\sup\{s\geq 0,X_{s}=0\}$:

• •

$\mathcal{Q}[g=\infty]=0$;

• •

For all $s\geq 0$, and for all $\mathcal{F}_{s}$-measurable, bounded random variables $F_{s}$,

[TABLE]

In this article, we prove that there is a link between penalisations and the measure $\mathcal{Q}$ involved in Theorem 1.8. More precisely, we show that if $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$ is the natural augmentation of the space $\mathcal{C}(\mathbb{R}_{+},\mathbb{R})$ endowed with its canonical filtration and the law of a diffusion satisfying some technical conditions, if $(X_{s})_{s\geq 0}$ is the canonical process, and if $(\Gamma_{t})_{t\geq 0}$ is a nonincreasing family of nonnegative random variables on the same space, tending to $\Gamma_{\infty}$ when $t$ goes to infinity, and such that

[TABLE]

and

[TABLE]

then under some precise assumptions stated below, the probability measure

[TABLE]

converges weakly (in the sense of penalisations) to the limit

[TABLE]

In the Brownian case, a similar penalisation result, involving the measure $\mathcal{Q}$, is obtained in [12].

The present paper is organized as follows: in Section 2, we detail the precise setting under which our penalisation result is available and we state precisely this result, in Section 3, we give some estimates of hitting times of the canonical diffusion, and in Section 4, we use these estimates in order to complete the proof of our main theorem.

2. The general setting

The general framework in which one can state our penalisation result was introduced by Salminen, Vallois and Yor in [16], in a slightly different and more general form, and it was also used by Najnudel, Roynette and Yor in [12], Chapter 3. Note that in both cases the filtration is not completed. In this present paper, we use the natural augmentation, which allows us to define the local time everywhere, and to apply Theorem 1.8.

The underlying filtered probability space is constructed as follows. Let $\Omega$ be the space of continuous functions from $\mathbb{R}_{+}$ to $\mathbb{R}_{+}$, $(\mathcal{F}^{0}_{s})_{s\geq 0}$, the natural filtration of $\Omega$ (not completed), and $\mathcal{F}^{0}$, the $\sigma$-algebra generated by $(\mathcal{F}^{0}_{s})_{s\geq 0}$. We assume that the following holds:

• •

the probability $\mathbb{P}^{0}$, defined on $(\Omega,\mathcal{F}^{0})$, is such that under $\mathbb{P}^{0}$, the canonical process is a recurrent diffusion in natural scale, starting from a fixed point $x_{0}\geq 0$, with zero as an instantaneously reflecting barrier;

• •

the speed measure $m(x)$ of this diffusion is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}_{+}$, with a continuous density $m:\mathbb{R}^{*}_{+}\rightarrow\mathbb{R}^{*}_{+}$;

• •

we assume that $m(x)$ is equivalent to $cx^{\beta}$ when $x$ goes to infinity, for some $c>0$ and $\beta>-1$, and we suppose that there exists $C>0$ such that for all $x>0$, $m(x)\leq Cx^{\beta}$ if $\beta\leq 0$, and $m(x)\leq C(1+x^{\beta})$ if $\beta>0$;

• •

the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$ is constructed as the natural augmentation of the space $(\Omega,\mathcal{F}^{0},(\mathcal{F}^{0}_{s})_{s\geq 0},\mathbb{P}^{0})$: $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$ satisfies the property (NP) and under $\mathbb{P}$, the law of the canonical process is a diffusion with the same parameters as under $\mathbb{P}^{0}$.

This diffusion is in natural scale and, as in [12], one can deduce that the canonical process $(X_{s})_{s\geq 0}$ is a submartingale of class $(\Sigma)$. In particular, $X_{s}$ is integrable for all $s\geq 0$, moreover, the local time $(L_{s})_{s\geq 0}$ of $(X_{s})_{s\geq 0}$ at level zero, is its increasing process. One deduces that Theorem 1.8 applies, and one can construct the corresponding measure $\mathcal{Q}$. Let us check that we are in the situation where $L_{\infty}=\infty$, $\mathbb{P}$-almost surely. Indeed, let $T^{(0)}$ be the first hitting time of zero by $(X_{s})_{s\geq 0}$, and for $n\geq 1$, let $T^{(n)}$ be the first hitting time of zero after $T^{(n-1)}+1$. The variables $(T^{(n)})_{n\geq 0}$ are stopping times, $\mathbb{P}$-almost surely finite since $(X_{s})_{s\geq 0}$ is recurrent. By the strong Markov property the variables $(L_{T^{(n)}}-L_{T^{(n-1)}})_{n\geq 1}$ are i.i.d., which implies that $L_{\infty}=0$ almost surely, or $L_{\infty}=\infty$ almost surely. Let us suppose that the first case holds. One deduces that $(X_{s})_{s\geq 0}$ is a martingale, and by the optional sampling theorem, for all $s,u\geq 0$,

[TABLE]

which implies:

[TABLE]

and then $X_{T^{(0)}+s}=0$ a.s. on the event $\{T^{(0)}\leq u\}$. Since $u$ can be arbitrarily chosen and $T^{(0)}<\infty$ a.s., $X_{T^{(0)}+s}=0$ a.s., which contradicts the fact that zero is an instantaneously reflecting barrier. The fact that $L_{\infty}=\infty$ implies that $g^{[a]}<\infty$ $\mathcal{Q}$-almost everywhere, for

[TABLE]

as proved in [10]. This property is an important element in the proof of our penalisation result, which can now be rigorously stated. The relevant class of functionals $(\Gamma_{t})_{t\geq 0}$ is defined as follows:

Definition 2.1**.**

Let us suppose that the assumptions of Theorem 1.8 are satisfied. We say that a process $(\Gamma_{t})_{t\geq 0}$ belongs to the class (C) if it is nonnegative, uniformly bounded, nonincreasing, càdlàg and adapted with respect to $(\mathcal{F}_{t})_{t\geq 0}$, if there exists $a>0$ such that for all $t\geq 0$, $\Gamma_{t}=\Gamma_{g^{[a]}}$ on the set $\{t\geq g^{[a]}\}$, and if the decreasing limit of $\Gamma_{t}$ at infinity, denoted $\Gamma_{\infty}$, is $\mathcal{Q}$-integrable.

This definition is stated in [10], and in a slightly different way in [12]. The main result of this article is the following:

Theorem 2.2**.**

We suppose that the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{s})_{s\geq 0},\mathbb{P})$ and the diffusion process $X$ are constructed as above. Let $\mathcal{Q}$ be the $\sigma$-finite measure associated with $X$ from Theorem 1.8 and let $(\Gamma_{t})_{t\geq 0}$ be a process in the class (C), such that $\mathcal{Q}[\Gamma_{\infty}]>0$. For all $t\geq 0$:

[TABLE]

and one can define a probability measure $\mathbb{Q}_{t}$ on $(\Omega,\mathcal{F})$ by

[TABLE]

Then the probability measure:

[TABLE]

is the weak limit of $\mathbb{Q}_{t}$ in the sense of penalisations, i.e. for all $s\geq 0$, and for all events $\Lambda_{s}\in\mathcal{F}_{s}$,

[TABLE]

The proof of Theorem 2.2 is given in Section 3 and Section 4.

Remark 2.3*.*

This penalisation result applies in particular for the power $2r$ of a Bessel process of dimension $2(1-r)$, if $0<r<1$. For $r=1$, the corresponding diffusion is a reflected Brownian motion, and our result is very similar to the result obtained for Brownian motion in [12]. The proof of Theorem 2.2 is given in Sections 3 and 4.

3. Estimates of hitting times

In order to prove Theorem 2.2, we need to estimate the distribution of the hitting times of the canonical process $(X_{s})_{s\geq 0}$ under $\mathbb{P}$. These estimates involve Sturm-Liouville equation in a crucial way, that is why we first prove the following lemma, giving some information about the solutions of this equation:

Lemma 3.1**.**

Let $A$ be a continuous function from $\mathbb{R}_{+}^{*}$ to $\mathbb{R}_{+}^{*}$, integrable in the neighborhood of zero, but not in the neighborhood of infinity. Then there exists a unique function $\Phi_{A}$ from $\mathbb{R}_{+}$ to $\mathbb{R}$, continuous on $\mathbb{R}_{+}$, bounded, twice differentiable on $\mathbb{R}_{+}^{*}$, such that $\Phi_{A}(0)=1$ and satisfying Sturm-Liouville equation:

[TABLE]

for all $t>0$. This function is strictly positive, decreasing to zero at infinity, and continuously differentiable everywhere in $\mathbb{R}_{+}$. Moreover, if $A_{1}$ and $A_{2}$ are two functions satisfying the same assumptions as $A$, and if $A_{1}(t)\geq A_{2}(t)$ for all $t>0$, then $|\Phi_{A_{1}}^{\prime}(0)|\geq|\Phi_{A_{2}}^{\prime}(0)|$.

Proof.

By Cauchy-Lipschitz theorem, for all $t_{0}>0$, $a,b\in\mathbb{R}$, there exists a unique maximal solution $\Theta_{t_{0},a,b}$ of (3.1) on an interval of the form $[t_{0},v)$ for some $v\in(t_{0},\infty]$, which satisfies $\Theta_{t_{0},a,b}(t_{0})=a$ and $\Theta^{\prime}_{t_{0},a,b}(t_{0})=b$. Since $A$ is locally bounded on $\mathbb{R}_{+}^{*}$, $\Theta_{t_{0},a,b}$ is in fact well-defined on $[t_{0},\infty)$. By linearity, one has:

[TABLE]

For all $t>t_{0}$, $\Theta_{t_{0},1,0}(t)$, $\Theta^{\prime}_{t_{0},1,0}(t)$, $\Theta_{t_{0},0,1}(t)$ and $\Theta^{\prime}_{t_{0},0,1}(t)$ are strictly positive, moreover, $\Theta_{t_{0},1,0}$ and $\Theta_{t_{0},0,1}$ tend to infinity at infinity. On the other hand, let us suppose that $\Theta_{t_{0},a,b}$ is not bounded from above (resp. from below). Then there exists $t>t_{0}$ such that $\Theta_{t_{0},a,b}(t)>0$ and $\Theta^{\prime}_{t_{0},a,b}(t)\geq 0$ (resp. $\Theta_{t_{0},a,b}(t)<0$ and $\Theta^{\prime}_{t_{0},a,b}(t)\leq 0$): one deduces that $\Theta_{t_{0},a,b}$ tends to infinity (resp. minus infinity) at infinity. Hence, for $b_{1}>b_{2}$, at least one of the two following holds:

• •

$\Theta_{t_{0},a,b_{1}}$ tends to infinity at infinity.

• •

$\Theta_{t_{0},a,b_{2}}$ tends to minus infinity at infinity.

Indeed, if none of these holds, then $\Theta_{t_{0},a,b_{1}}$ is bounded from above, and $\Theta_{t_{0},a,b_{2}}$ is bounded from below, which implies that $\Theta_{t_{0},0,1}$ is bounded from above: a contradiction. Now, for $b$ large enough, $\Theta_{t_{0},a,b}(t_{0}+1)$ and $\Theta^{\prime}_{t_{0},a,b}(t_{0}+1)$ are strictly positive, which implies that $\Theta_{t_{0},a,b}$ tends to infinity at infinity, similarly, for $b$ small enough, $\Theta_{t_{0},a,b}$ tends to minus infinity at infinity. One deduces that there exists $b(t_{0},a)$ such that for $b>b(t_{0},a)$, $\Theta_{t_{0},a,b}$ tends to infinity and for $b<b(t_{0},a)$, $\Theta_{t_{0},a,b}$ tends to minus infinity. Let us suppose that for some $t\geq t_{0}$, $\Theta_{t_{0},a,b(t_{0},a)}(t)$ and $\Theta^{\prime}_{t_{0},a,b(t_{0},a)}(t)$ have the same sign, and that at least one of them is different from zero. Since one can increase $t$ by a small quantity, one can suppose that $\Theta_{t_{0},a,b(t_{0},a)}(t)$ and $\Theta^{\prime}_{t_{0},a,b(t_{0},a)}(t)$ have strictly the same sign (recall that $\Theta_{t_{0},a,b(t_{0},a)}(t)>0$ implies that $\Theta^{\prime\prime}_{t_{0},a,b(t_{0},a)}(t)>0$). On deduces that for $b$ sufficiently close to $b(t_{0},a)$, $\Theta_{t_{0},a,b}(t)$ and $\Theta^{\prime}_{t_{0},a,b}(t)$ have the same sign, independent of $b$. Therefore, $\Theta_{t_{0},a,b}$ and $\Theta^{\prime}_{t_{0},a,b}$ tend to a limit equal to infinity or minus infinity, independently of $b$: this is a contradiction. At this step, we know that for all $t\geq t_{0}$, $\Theta_{t_{0},a,b(t_{0},a)}(t)=\Theta^{\prime}_{t_{0},a,b(t_{0},a)}(t)=0$ or $\Theta_{t_{0},a,b(t_{0},a)}(t)\Theta^{\prime}_{t_{0},a,b(t_{0},a)}(t)<0$. If the first case holds for some $t\geq t_{0}$, by using Cauchy-Lipschitz theorem in both directions of the time, we deduce that $a=b(t_{0},a)=0$ and $\Theta_{t_{0},a,b(t_{0},a)}$ is identically zero. Otherwise, $\Theta^{2}_{t_{0},a,b(t_{0},a)}$ is strictly decreasing, which implies that it remains strictly positive, hence, $\Theta_{t_{0},a,b(t_{0},a)}$ is strictly positive and strictly decreasing for $a>0$, strictly negative and strictly incrasing for $a<0$. In particular, it converges almost surely to a limit $l$. If $l>0$, then for all $t\geq t_{0}$, $\Theta_{t_{0},a,b(t_{0},a)}(t)\geq l$ and $\Theta^{\prime\prime}_{t_{0},a,b(t_{0},a)}(t)\geq lA(t)$, which implies that $\Theta^{\prime}_{t_{0},a,b(t_{0},a)}$ tends to infinity at infinity, since $A$ is not integrable at infinity: this is impossible. Since $l<0$ gives also a contradiction, one has $l=0$. To summarize, we have proved that for all $t_{0}>0$ and all $a\in\mathbb{R}$, there exists a unique solution $\Theta_{t_{0},a}$ of (3.1) defined on $[t_{0},\infty)$, equal to $a$ at $t_{0}$, and which does not tend to infinity or minus infinity at infinity. This solution is identically zero if $a=0$, it is strictly monotone and tends to zero at infinity, if $a\neq 0$. In fact, by linearity, one has $\Theta_{t_{0},a}=a\Theta_{t_{0},1}$ for all $a\in\mathbb{R}$. The uniqueness of $\Theta_{t_{0},a}$ implies that for $t_{0}>t_{1}>0$, $a\in\mathbb{R}$, the restriction of $\Theta_{t_{1},a}$ to $[t_{0},\infty)$ is equal to $\Theta_{t_{0},b}$ where $b=\Theta_{t_{1},a}(t_{0})$. This compatibility implies that if we define $\Theta$ from $\mathbb{R}_{+}^{*}$ to $\mathbb{R}_{+}^{*}$ by:

[TABLE]

for $t\geq 1$, and

[TABLE]

for $t<1$, then $\Theta$ is a solution of (3.1), strictly decreasing and tending to zero at infinity. Now, since $|\Theta^{\prime}|$ is decreasing, one has, for all $t\in(0,1)$,

[TABLE]

for $L:=1+1/|\Theta^{\prime}(1)|$. One deduces that

[TABLE]

and then

[TABLE]

Since $A$ is integrable in the neighborhood of zero, $|\Theta^{\prime}(t)|$, and then $\Theta(t)$, is bounded for $t\in(0,1)$. Since $\Theta$ is decreasing, it can be defined at zero by continuity. Moreover, $\Theta^{\prime}(t)$ converges when $t$ goes to zero, and one can check that its limit is equal to the derivative at zero of the extension of $\Theta$. One then deduces the existence and the properties of regularity of $\Phi_{A}$ by setting, for all $t\geq 0$:

[TABLE]

To prove uniqueness, let us suppose that $\Phi_{1}$ and $\Phi_{2}$ satisfy the conditions given in Lemma 3.1. The functions $\Phi_{1}$ and $\Phi_{2}$ cannot vanish on $\mathbb{R}_{+}^{*}$: otherwise, by the properties of the solutions of (3.1) on the intervals of the form $[t_{0},\infty)$ for $t_{0}>0$, they would be identically zero. Then $\Phi_{1}$, $\Phi_{2}$ are strictly positive, and they are proportional to each other on every interval of the form $[t_{0},\infty)$ for $t_{0}>0$. In other words, $\Phi_{1}/\Phi_{2}$ is constant on $[t_{0},\infty)$ for all $t_{0}>0$, and then on $\mathbb{R}^{*}$, finally, on $\mathbb{R}_{+}$ by continuity at zero. Since $\Phi_{1}(0)=\Phi_{2}(0)=1$, $\Phi_{1}=\Phi_{2}$ everywhere. Let us now suppose that $A_{1}\geq A_{2}$ satisfy the assumptions of Proposition 3.1, and that $|\Phi^{\prime}_{A_{1}}(0)|<|\Phi^{\prime}_{A_{2}}(0)|$. One has $\Phi^{\prime}_{A_{1}}(0)>\Phi^{\prime}_{A_{2}}(0)$. Let $T>0$ be the infimum of the times $t$ such that $\Phi^{\prime}_{A_{1}}(t)\leq\Phi^{\prime}_{A_{2}}(t)$. On the interval $(0,T)$, one has $\Phi^{\prime}_{A_{1}}>\Phi^{\prime}_{A_{2}}$, and then $\Phi_{A_{1}}\geq\Phi_{A_{2}}$, since $\Phi_{A_{1}}(0)=\Phi_{A_{2}}(0)$. Now, since $A_{1}\geq A_{2}$, one deduces that

[TABLE]

which implies:

[TABLE]

If $T$ is supposed to be finite, one deduces a contradiction at time $T$. Hence $T$ is infinite and 3.2 holds everywhere. One deduces that

[TABLE]

for all $t\geq 0$, and then $\Phi_{A_{1}}-\Phi_{A_{2}}$ tends to infinity at infinity, which is absurd. ∎

Once Lemma 3.1 is proved, one can state and show the results on hitting times which are involved in the proof of Theorem 2.2. In the sequel of the paper, for any $x\geq 0$, we denote by $\mathbb{P}_{x}$ the distribution of a diffusion which starts at $x$, with the same parameters as the canonical process under $\mathbb{P}$. In particular, $\mathbb{P}_{x_{0}}=\mathbb{P}$. With this notiation, one has the following lemma:

Lemma 3.2**.**

For all $\lambda>0$, there exists a unique function $\Phi_{\lambda}$ such that $\Phi_{\lambda}(0)=1$ and for all $x\geq y\geq 0$:

[TABLE]

where $T_{y}$ is the first hitting time of $y$ by the canonical process, and $\alpha:=1/(\beta+2)$ lies in the interval $(0,1)$. Moreover, there exists $\lambda_{0}>0$ such that the following properties hold:

• •

$\Phi_{\lambda}(t)$* converges to one when $(\lambda,t)$ goes to zero.*

• •

$\Phi_{\lambda}(t)$* converges to zero when $t$ goes to infinity, uniformly in $\lambda\in(0,\lambda_{0})$.*

• •

For all $\lambda>0$, $\Phi_{\lambda}$ is continuously differentiable everywhere in $\mathbb{R}_{+}$.

• •

The map: $(\lambda,t)\longrightarrow\Phi^{\prime}_{\lambda}(t)$ is unformly bounded on $(0,\lambda_{0})\times\mathbb{R}_{+}$.

• •

$|\Phi^{\prime}_{\lambda}(t)|$* converges to a constant $K>0$ when $(\lambda,t)$ goes to zero.*

Proof.

Let us define, for all $u\geq 0$:

[TABLE]

By the strong Markov property one has

[TABLE]

and then

[TABLE]

where for $t\geq 0$,

[TABLE]

One deduces the existence of $\Phi_{\lambda}$, its uniqueness is clear. Now, by classical properties of diffusions (see [4] for example), the functions $\Psi_{\lambda}$, and then $\Phi_{\lambda}$, are continuous on $\mathbb{R}_{+}$, twice differentiable on $\mathbb{R}_{+}^{*}$ and $\Psi_{\lambda}$ satisfies:

[TABLE]

where $\mathcal{G}$ is the infinitesimal generator of the diffusion $X$. One deduces:

[TABLE]

for

[TABLE]

By assumption, $p(u)$ is strictly positive, continuous with respect to $u>0$, it tends to $c$ when $u$ goes to infinity, and for all $u>0$,

[TABLE]

One deduces that

[TABLE]

satisfies the assumptions of Lemma 3.1, moreover, with the notation of this Lemma:

[TABLE]

in particular, this function is continuously differentiable everywhere in $\mathbb{R}_{+}$ (third item of Lemma 3.2). Now, let us define the functions:

[TABLE]

and for $\epsilon>0$, $\lambda\geq 0$:

[TABLE]

Since $p$ tends to $c$ at infinity, for all $\epsilon>0$, there exists $\lambda(\epsilon)\in(0,1)$ such that for all $\lambda\in[0,\lambda(\epsilon)]$:

[TABLE]

All these functions satisfy the conditions of Lemma 3.1, and one deduces:

[TABLE]

Now, by checking Sturm-Liouville equation, one sees that for all $t\geq 0$:

[TABLE]

[TABLE]

and

[TABLE]

Therefore:

[TABLE]

where

[TABLE]

and

[TABLE]

do not depend on $\lambda\leq\lambda(\epsilon)$ and tend to $K:=|\Phi^{\prime}_{A_{0}}(0)|$ when $\epsilon$ goes to zero. One deduces that

[TABLE]

and

[TABLE]

Since one supposes $\lambda\leq\lambda(\epsilon)\leq 1$, one deduces that for all $t\geq 0$:

[TABLE]

where

[TABLE]

decreases to zero when $\epsilon$ goes to zero. In other words:

[TABLE]

where $K_{3}(\epsilon)<\infty$ decreases to $K$ when $\epsilon$ goes to zero. In particular, for $\lambda\leq\lambda(1)$:

[TABLE]

which proves the fourth item of Lemma 3.2. Moreover, for all $\epsilon>0$:

[TABLE]

and then, by taking $\epsilon$ going to zero,

[TABLE]

On the other hand, again for $\lambda\in[0,\lambda(\epsilon)]$, one has

[TABLE]

and then

[TABLE]

where $K_{4}(\epsilon)$ tends to $K$ when $\epsilon$ tends to zero. We deduce that

[TABLE]

for all $t\leq\epsilon$, and then

[TABLE]

This implies, by letting $\epsilon$ go to zero that

[TABLE]

The inequalities (3.3) and (3.4) imply the fifth item of Lemma 3.2. Moreover, for all $\lambda\geq 0$ and $t\geq 0$:

[TABLE]

Hence, for $\lambda\leq\lambda(1)$:

[TABLE]

Since $tK_{3}(1)$ tends to zero when $(\lambda,t)\in[0,\lambda(1)]\times\mathbb{R}_{+}$ tends to zero, one deduces the first item of Lemma 3.2. It now remains to prove the second item. Let us suppose that $\lambda\leq 1$ and $t\geq 1$. In this case:

[TABLE]

where

[TABLE]

One deduces that

[TABLE]

or

[TABLE]

By integrating and taking the exponential, one obtains

[TABLE]

which implies, for all $\lambda\leq 1$ and $t\geq 1$:

[TABLE]

where

[TABLE]

We can remark that this upper bound trivially holds for $t<1$, since $\theta(t)$ is greater than or equal to one in this case. Now, $\theta(t)$ tends to zero when $t$ goes to infinity (since $\Phi_{A_{0}}$ tends to zero), independently of $\lambda\leq 1$, which completes the proof of Lemma 3.2. ∎

The following lemma gives some information about the marginal distributions of the canonical process under $\mathbb{P}_{x}$:

Lemma 3.3**.**

One can define a function $\chi$ from $\mathbb{R}_{+}^{*}$ to $\mathbb{R}_{+}$, tending to zero at zero, and satisfying the following property: for all $u>0$, there exists $t_{0}(u)>0$ such that for all $x\geq 0$, $t\geq t_{0}(u)$:

[TABLE]

Proof.

Let us fix $u>0$. By coupling, one obtains immediately, for all $x\geq 0$, $t>0$,

[TABLE]

Now one has (see, for example, [12], p. 89):

[TABLE]

where $p$ is the density of the semi-group of $(X_{t})_{t\geq 0}$. Again by coupling, one sees that $p(t,y,0)\leq p(t,0,0)$, which implies that

[TABLE]

Now, if $t\geq u^{-1/\alpha}$, one has necessarily $t^{\alpha}u\leq(t^{\alpha}u)^{\beta+1}$ for $\beta>0$, which implies:

[TABLE]

where

[TABLE]

Hence, we are done, provided that there exists $L_{1}>0$ such that $p(t,0,0)\leq L_{1}\,t^{\alpha-1}$ for $t$ large enough. If $t>t^{\prime}>0$, one has

[TABLE]

which implies that $p(t,0,0)$ is decreasing with respect to $t$. One deduces:

[TABLE]

where for $\lambda>0$,

[TABLE]

Now from a result by Salminen, Vallois and Yor in [16] (p. 5, equation just after (3)), one obtains

[TABLE]

which implies, with the notation of Lemma 3.2, that

[TABLE]

Therefore we only need to prove that for some $L_{2}>0$, and for $t$ large enough

[TABLE]

Now, from Lemma 3.2 (first item), we know that there exist $t_{0},v_{0}>0$ such that for all $t\geq t_{0}$ and all $v\leq v_{0}$, $\Phi_{1/t}(v)\geq 1/2$. One deduces that for all $t\geq t_{0}\vee v_{0}^{-1/\alpha}$, $v_{0}t^{\alpha}\geq 1$:

[TABLE]

where $\delta$ is the infimum of $y^{-\beta}m(y)$ on $[1/2,\infty)$, strictly positive since $m(y)$ is strictly positive, continuous and equivalent to $cy^{\beta}$ at infinity. ∎

From Lemma 3.3, one deduces the following result, which majorizes the probability, for the canonical process, to hit $[0,a]$ during a small interval of time:

Lemma 3.4**.**

One can define a function $\rho$ from $(0,1)$ to $\mathbb{R}_{+}^{*}$, tending to zero at zero, and satifying the following property: for all $a\geq 0$, $\gamma\in(0,1)$, there exists $t_{0}(a,\gamma)$ such that for every $x\geq 0$, $t\geq t_{0}(a,\gamma)$:

[TABLE]

Proof.

Let us fix $a$ and $\gamma$. One has immediately:

[TABLE]

where, for $y\leq a$:

[TABLE]

and for $y>a$:

[TABLE]

In any case, one has:

[TABLE]

Now, there exist $\lambda_{1},v_{1}>0$ such that $\Phi_{\lambda}(v)\geq e/3$ for all $\lambda\leq\lambda_{1}$ and $v\leq v_{1}$ (first item of Lemma 3.2). Hence, there exists $t_{1}>0$ (depending on $\gamma$ and $a$) such that for $t\geq t_{1}$:

[TABLE]

and then for all $y\geq 0$:

[TABLE]

By the second item of Lemma 3.2, there exists a function $q$, bounded by one, decreasing to zero at infinity, such that for all $\lambda\leq\lambda_{0}$, $v>0$:

[TABLE]

Hence, if $t\geq t_{1}\vee 1/(\lambda_{0}\gamma)$, for all $y\geq 0$,

[TABLE]

which implies:

[TABLE]

and then, for any $v\geq 0$:

[TABLE]

We now fix

[TABLE]

One deduces, by Lemma 3.3, that there exists $t_{2}\geq 0$ (depending on $a$ and $\gamma$), such that for $t\geq t_{2}$:

[TABLE]

which tends to zero at zero. ∎

We have now all the estimates of hitting times needed in the proof of Theorem 2.2, which is finished in Section 4.

4. Proof of the main theorem

In order to prove Theorem 2.2, we essentially need to estimate the behaviour of the expectation of $\Gamma_{t}$ under $\mathbb{P}$, when $t$ goes to infinity. This expectation will be implicitly splitted as follows:

[TABLE]

where $s\geq 0$, $\gamma\in(0,1/2)$ and

[TABLE]

(recall that $g^{[a]}$ denotes the supremum of $u\in\mathbb{R}_{+}$ such that $X_{u}\leq a$). Moreover, we shall use a Tauberian theorem: this is the reason why we assume that the speed measure $m(x)$ of the diffusion $(X_{s})_{s\geq 0}$ behaves like a power of $x$ at infinity. The proof of Theorem 2.2 is divided into several steps, each of them corresponding to a lemma or a proposition. The first step is the following:

Lemma 4.1**.**

For all $a>0$, $s>0$, $u\geq 0$, and for all bounded, $\mathcal{F}_{s}$-measurable, nonnegative functionals $\Gamma_{s}$:

[TABLE]

where $K$ is the constant introduced in the last item of Lemma 3.2.

Proof.

One has by the Markov property that

[TABLE]

Now, by last item of Lemma 3.2, for $v$, $a$, $X_{s}$ fixed:

[TABLE]

Moreover, by the fourth item, for $\lambda\leq\lambda_{0}$:

[TABLE]

where $K^{\prime}>0$ does not depend on $\lambda$, $v$, $a$ or $X_{s}$. Since $\mathbb{P}[\Gamma_{s}(X_{s}-a)_{+}]$ is dominated by $\mathbb{P}[X_{s}]<\infty$, one can apply dominated convergence, which yields

[TABLE]

and Lemma 4.1 for $u=s$. Now, if $M>0$ majorizes uniformly $\Gamma_{s}$, we have

[TABLE]

which tends to zero with $\lambda$. ∎

The next step is the following:

Lemma 4.2**.**

Let $a>0$ be fixed. Then, there exist $u_{0}>0$, $L>0$ such that for all $r>0$, $u\geq u_{0}$ and for all $\mathcal{F}_{r}$-measurable, bounded, nonnegative functionals $\Gamma_{r}$:

[TABLE]

Proof.

One has:

[TABLE]

From the fourth item of Lemma 3.2, one deduces that $|\Phi^{\prime}_{1/u}|$ is uniformly bounded by a constant $L^{\prime}$ for $u\geq 1/\lambda_{0}.$ Under these assumptions:

[TABLE]

for $u$ sufficiently large in order to make sure that $\Phi_{1/u}(au^{-\alpha})\geq 2/3$. ∎

Now, let us prove the following:

Lemma 4.3**.**

Let $a>0$ and let $(\Gamma_{t})_{t\geq 0}$ be a càdlàg, adapted, nonnegative, uniformly bounded and nonincreasing process, such that for all $t\geq 0$, $X_{t}=X_{g^{[a]}}$ on the set $\{t\geq g^{[a]}\}$. We define $\Gamma_{\infty}$ as the limit of $\Gamma_{t}$ for $t$ going to infinity (in particular, $\Gamma_{\infty}=\Gamma_{g^{[a]}}$ for $g^{[a]}<\infty$), and we suppose that $\Gamma_{\infty}$ is integrable with respect to $\mathcal{Q}$. Then, for all $\gamma\in(0,1/2)$, there exists $R>0$ such that for all $s\geq 0$:

[TABLE]

Proof.

For all $t>0$:

[TABLE]

Remark that the quantities involved here are equal to zero for $t\leq s/(1-\gamma)$. By Lemma 4.2, one deduces that for $t\geq u_{0}/\gamma$:

[TABLE]

Now, if $g^{[a]}\leq(1-\gamma)t$, $\Gamma_{(1-\gamma)t}=\Gamma_{\infty}$, which implies:

[TABLE]

One now deduces:

[TABLE]

Since

[TABLE]

we are done. ∎

Lemma 4.4**.**

Let $a>0$ and let $(\Gamma_{t})_{t\geq 0}$ be a càdlàg, adapted, nonnegative, uniformly bounded and nonincreasing process, such that for all $t\geq 0$, $X_{t}=X_{g^{[a]}}$ on the set $\{t\geq g^{[a]}\}$. We define $\Gamma_{\infty}$ as the limit of $\Gamma_{t}$ for $t$ going to infinity, and we suppose that $\Gamma_{\infty}$ is integrable with respect to $\mathcal{Q}$. Then, for all $\gamma\in(0,1/2)$, for all $s\geq 0$ and for all events $\Lambda_{s}\in\mathcal{F}_{s}$:

[TABLE]

Proof.

Let $v>s$. For $t>v$:

[TABLE]

By Lemma 4.1, one has

[TABLE]

which implies

[TABLE]

for all $w\geq 0$. Now, one also has:

[TABLE]

which implies:

[TABLE]

On the other hand:

[TABLE]

which implies by Lemma 4.3 that

[TABLE]

By comparing (4.1) and (4.2) and by taking $v\rightarrow\infty$, we are done, since $\Gamma_{\infty}$ is $\mathcal{Q}$-integrable and $g^{[a]}<\infty$, $\mathcal{Q}$-almost everywhere. ∎

The following result gives, for all $s\geq 0$ and all events $\Lambda_{s}\in\mathcal{F}_{s}$, an equivalent of the expectation of $\Gamma_{t}\mathds{1}_{\Lambda_{s}}$ under $\mathbb{P}$, for $t$ going to infinity, if $(\Gamma_{t})_{t\geq 0}$ belongs to the class (C).

Proposition 4.5**.**

Let $(\Gamma_{t})_{t\geq 0}$ be a process in the class (C). Then, there exists $D>0$ such that for all $s\geq 0$ and for all events $\Lambda_{s}\in\mathcal{F}_{s}$:

[TABLE]

Proof.

Lemma 4.4 implies immediately (by taking any $\gamma$) that

[TABLE]

Now, for $\lambda>0$, let us define:

[TABLE]

and for $\lambda>0$, $\gamma<(0,1/2)$,

[TABLE]

and

[TABLE]

Here $a>0$ is chosen in order to have $\Gamma_{t}=\Gamma_{g_{t}^{[a]}}$ on the event $\{t\geq g_{t}^{[a]}\}$. For all $t\geq 0$, by the Markov property

[TABLE]

By Lemma 3.4, for $t\geq 2t_{0}(a,2\gamma)$:

[TABLE]

If $M>0$ majorizes uniformly $\Gamma_{t}$ for all $t\geq 0$, one deduces:

[TABLE]

which implies:

[TABLE]

Therefore:

[TABLE]

which implies

[TABLE]

and finally

[TABLE]

for $\gamma$ such that $\rho(2\gamma)<1/2$ (this condition is always satisfied if $\gamma$ is small enough). Since, by Lemma 4.4,

[TABLE]

one deduces:

[TABLE]

and then,

[TABLE]

By making $\gamma\rightarrow 0$, one deduces:

[TABLE]

By taking (4.3) and (4.4) together and by using Tauberian theorem (recall that $\Gamma_{t}$ is decreasing with respect to $t$), we are done. ∎

Now, Theorem 2.2 can be deduced from Proposition 4.5 in a very simple way, as follows. Let us suppose that for some $t\geq 0$, $\Gamma_{t}=0$ almost surely with respect to $\mathbb{P}$. Since all the $\mathbb{P}$-negligible events in $\mathcal{F}_{t}$ are also $\mathcal{Q}$-negligible, one has $\mathcal{Q}[\Gamma_{t}]=0$, which contradicts the fact that $\mathcal{Q}[\Gamma_{\infty}]>0$, since $\Gamma_{t}$ decreases with respect to $t$. Then, $\mathbb{P}[\Gamma_{t}]>0$ for all $t\geq 0$, and obviously $\mathbb{P}[\Gamma_{t}]<\infty$, since $\Gamma_{t}$ is uniformly bounded. Hence, $\mathbb{Q}_{t}$ is well-defined. Moreover, for all $s\geq 0$ and $\Lambda_{s}\in\mathcal{F}_{s}$:

[TABLE]

Now by Proposition 4.5,

[TABLE]

and

[TABLE]

Therefore:

[TABLE]

which completes the proof of Theorem 2.2.