Spectral Heat Content for L\'evy Processes

TL;DR

This paper investigates the asymptotic behavior of spectral heat content for various Lévy processes, revealing stability under perturbations and extending results to different process variations and set types.

Contribution

It provides new asymptotic results for spectral heat content of Lévy processes of bounded and unbounded variation, including stability under Lévy measure perturbations.

Findings

Asymptotic behavior characterized for bounded variation Lévy processes.

Spectral heat content analyzed for open sets of finite measure in one dimension.

Stability of asymptotics under Lévy measure perturbations established.

Abstract

In this paper we study the spectral heat content for various L\'evy processes. We establish the asymptotic behavior of the spectral heat content for L\'{e}vy processes of bounded variation in $\mathbb{R}^{d}$, $d\geq 1$. We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in $\mathbb{R}$ with respect to L\'{e}vy processes of unbounded variation under certain conditions on their characteristic exponents. Finally we establish that the asymptotic behavior of the spectral heat content is stable under integrable perturbations to the L\'{e}vy measure.

Equations

$$H_{\Omega }(t)={\int }_{\Omega }{\mathbb{P}}_x(X_t\in \Omega )\mathrm{d}x,$$

$$Q_{\Omega }(t):={\int }_{\Omega }{\mathbb{P}}_x({\tau }_{\Omega }>t)\mathrm{d}x,$$

$$\psi (\xi )=⟨\xi ,A\xi ⟩-i⟨\xi ,\gamma ⟩-{\int }_{{\mathbb{R}}^d}\left(e^{i⟨\xi ,y⟩}-1-i⟨\xi ,y⟩1_{\{\Vert y\Vert \le 1\}}\right)\nu (\mathrm{d}y),$$

$$\nu (\{0\})=0\mathrm{and}{\int }_{{\mathbb{R}}^d}\min \{1,{\Vert y\Vert }^2\}\nu (\mathrm{d}y)<\infty .$$

$$A=0\mathrm{and}{\int }_{\Vert y\Vert \le 1}\Vert y\Vert \nu (\mathrm{d}y)<\infty .$$

$$\psi (\xi )=i⟨\xi ,{\gamma }_0⟩+{\int }_{{\mathbb{R}}^d}\left(1-e^{i⟨\xi ,y⟩}\right)\nu (\mathrm{d}y),$$

$$\displaystyle h(r)={\lVert A\rVert}r^{-2}+r^{-1}\Big{\lvert}\gamma+\int_{\mathbb{R}^{d}}y\left(1_{\{{\lVert y\rVert}<r\}}-1_{\{{\lVert y\rVert}<1\}}\right)\nu(\mathrm{d}y)\Big{\rvert}+\int_{\mathbb{R}^{d}}\min\{1,{\lVert y\rVert}^{2}r^{-2}\}\,\nu(\mathrm{d}y).$$

$$\mathbb{P}(\underset{s\le t}{\sup }|X_s|>r)\le C_{1}th(r).$$

$$\mathrm{Per}(\Omega )=\sup \left\{{\int }_{{\mathbb{R}}^d}{1}_{\Omega }(x)\mathrm{div}\varphi (x)\mathrm{d}x:\varphi \in C_c^1({\mathbb{R}}^d,{\mathbb{R}}^d),{\Vert \varphi \Vert }_{\infty }\le 1\right\}.$$

$$\mathrm{Per}(\Omega )={\pi }^{1/2}\underset{t\to 0}{\lim }{t}^{-1/2}{\int }_{\Omega }{\int }_{{\Omega }^c}{p}_t^{(2)}(x,y)\mathrm{d}y\mathrm{d}x,$$

$$p_t^{(2)}(x,y)=(4\pi t{)}^{-d/2}{e}^{-{\Vert x-y\Vert }^2/4t}$$

$${\mathrm{Per}}_{\mathbf{X}}(\Omega )={\int }_{\Omega }{\int }_{{\Omega }^c-x}\nu (\mathrm{d}y)\mathrm{d}x.$$

$${\mathrm{Per}}_{{\mathbf{S}}^{(\alpha )}}(\Omega ):={\int }_{\Omega }{\int }_{{\Omega }^c}\frac{c(d,\alpha )\mathrm{d}y\mathrm{d}x}{{\Vert x-y\Vert }^{d+\alpha }},$$

$$c(d,\alpha ):=\frac{\alpha \Gamma (\frac{d+\alpha }{2})}{2^{1-\alpha }{\pi }^{d/2}\Gamma (1-\frac{\alpha }{2})}.$$

$$V(f,G)=\sup \{{\int }_{G}f(x)\text{div}\phi (x)\mathrm{d}x:\phi \in C_c^1(G,{\mathbb{R}}^d),\Vert \phi {\Vert }_{\infty }\le 1\}.$$

$$V_u(f,G)=\sup \{{\int }_{G}f(x)⟨\nabla \phi (x),u⟩\mathrm{d}x:\phi \in C_c^1(G,{\mathbb{R}}^d),\Vert \phi {\Vert }_{\infty }\le 1\}.$$

$$g_{\Omega }(y)=|\Omega \cap (\Omega +y)|={\int }_{{\mathbb{R}}^d}{1}_{\Omega }(x)1_{\Omega +y}(x)\mathrm{d}x={\int }_{{\mathbb{R}}^d}{1}_{\Omega }(x)1_{\Omega }(x-y)\mathrm{d}x.$$

$$g_{\Omega }(y)\le g_{\Omega }(0)=|\Omega |\text{ and }{g}_{\Omega }(-y)=g_{\Omega }(y).$$

$$f_{\Omega }(y):=g_{\Omega }(0)-g_{\Omega }(y).$$

$${\mathrm{Per}}_{\mathbf{X}}(\Omega )={\int }_{{\mathbb{R}}^d}{f}_{\Omega }(y)\nu (\mathrm{d}y).$$

$$\underset{|y|\to 0}{\lim }{f}_{\Omega }(y)=0.$$

$$\underset{|y|\to 0}{\lim }{1}_{\Omega }(x)1_{{\Omega }^c}(x-y)=0.$$

$$\left|g_{\Omega }(x)-g_{\Omega }(y)\right|\le g_{\Omega }(0)-g_{\Omega }(x-y),x,y\in {\mathbb{R}}^d.$$

$$\underset{r\to \infty }{\lim }\frac{f(\lambda r)}{f(r)}={\lambda }^{\alpha }.$$

$$\mathcal{L}f(x)=\underset{t\to 0}{\lim }\frac{P_{t}f(x)-f(x)}{t},$$

$$\mathcal{L}f(x)$$

$${\int }_{{\mathbb{R}}^d}{f}_{\Omega }(y)\nu (\mathrm{d}y)<\infty ,$$

$$\mathcal{L}{g}_{\Omega }(x)={\int }_{{\mathbb{R}}^d}\left(g_{\Omega }(x+y)-g_{\Omega }(x)\right)\nu (\mathrm{d}y).$$

$$|g_{\Omega }^{\epsilon }(x)-g_{\Omega }(x)|=\left|{\int }_{{\mathbb{R}}^d}{\varphi }_{\epsilon }(y)\left(g_{\Omega }(x-y)-g_{\Omega }(x)\right)\mathrm{d}y\right|$$

$$\underset{\epsilon \downarrow 0}{\lim }||g_{\Omega }^{\epsilon }-g_{\Omega }|{|}_{\infty }=0.$$

Full text

T. Grzywny was supported in part by National Science Centre (Poland): grant 2016/23/B/ST1/01665. R. Song is partially supported by a grant from the Simons Foundation (#429343, Renming Song).

1. Introduction

Let $\mathbf{X}=(X_{t})_{t\geq 0}$ be a Lévy process in $\mathbb{R}^{d}$. For any open set $\Omega\subset\mathbb{R}^{d}$, the (regular) heat content of $\Omega$ with respect to $\mathbf{X}$ is defined to be

[TABLE]

while the spectral heat content of $\Omega$ with respect to $\mathbf{X}$ is defined to be

[TABLE]

where $\tau_{\Omega}=\inf\{t>0:X_{t}\in\Omega^{c}\}$ is the first time the process $\mathbf{X}$ exits $\Omega$.

The asymptotic behavior of the heat content and the spectral heat content has been studied intensively in the case of Brownian motion, see [20] and [24]–[29]. Recently significant progress has also been made in studying the heat content and the spectral heat content with respect to Lévy processes, see [1, 2, 3, 8, 12]. The asymptotic behavior of the heat content and the spectral heat content with respect to symmetric stable processes was studied in [1, 2, 3]. In particular, the exact asymptotic behavior of the spectral heat content of bounded open intervals with respect to symmetric stable processes in $\mathbb{R}$ was established in [3]. The asymptotic behavior of the heat content with respect to general Lévy processes was studied in [8], see also [9] for a generalization. In [12], an asymptotic expansion of the heat content with respect to some isotropic compound Poisson processes with compactly supported jumping kernels was established.

The purpose of this paper is to investigate the asymptotic behavior of the spectral heat content of general Lévy processes and generalize the results of [3] in several directions.

The organization of this paper is as follows. In Section 2 we recall some notions and present some preliminaries. In Section 3 we first study the heat content with respect to Lévy processes of bounded variation. In Theorem 4, we extend [8, Theorem 3] to any open set of finite Lebesgue measure and relax the finite perimeter condition. Then we use this to establish the asymptotic behavior of the spectral heat content for the same processes in Theorems 5 and 6. In Section 4 we investigate the asymptotic behavior of the spectral heat content with respect to Lévy processes of unbounded variation in $\mathbb{R}$. In this section we deal with two cases separately. In the first case, we assume that the characteristic exponent $\psi(\xi)$ of $\mathbf{X}$ is regularly varying of index $\alpha$ at infinity for some $\alpha\in(1,2]$. In the second case, we assume that $\mathbf{X}$ is a symmetric 1-stable process, that is, a Cauchy process. The main results in Section 4 are Theorems 10 and 21, where we establish the exact asymptotic behavior of the spectral heat content with respect to such processes. We note here that the asymptotic behavior of $Q_{\Omega}(t)$ depends on the geometry of $\Omega$. When $\alpha\in(1,2]$, both the number of adjacent components and the number of non-adjacent components matter, while in the case $\alpha=1$, only the number of non-adjacent components matters since the process can not hit a single point upon exiting the open set. Two components of $\Omega$ are said to be adjacent if the distance between them is zero. In Section 5 we study the stability of the spectral heat content. We prove in Theorem 22 that the asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measures. In Section 6 we give some examples where one can apply the results of this paper to get the asymptotic behavior of the spectral heat content.

2. Preliminaries

Let $\mathbf{X}=(X_{t})_{t\geq 0}$ be a Lévy process in $\mathbb{R}^{d}$. We denote by $(P_{t})$ the semigroup of $\mathbf{X}$ and by $\widehat{P}_{t}$ the adjoint operator of $P_{t}$. The characteristic exponent $\psi(\xi)$, $\xi\in\mathbb{R}^{d}$, of $\mathbf{X}$ is given by

[TABLE]

where $A$ is a symmetric non-negative definite $d\times d$ matrix, $\gamma\in\mathbb{R}^{d}$ and $\nu$ is a Lévy measure, that is

[TABLE]

The Lévy process $\mathbf{X}$ is of bounded variation (see [22, Theorem 21.9]) if and only if

[TABLE]

In this case the characteristic exponent has the following simple form

[TABLE]

where $\gamma_{0}=\int_{{\lVert y\rVert}\leq 1}y\,\nu(\mathrm{d}y)-\gamma.$ For a Lévy process of bounded variation, the quantity $\gamma_{0}$ defined above is called the drift of the process.

We introduce the following function related to $\mathbf{X}$, see [21]. For any $r>0$,

[TABLE]

Recall that there exists $C_{1}=C_{1}(d)>0$ (see [21, page 941]) such that

[TABLE]

Following [4, Section 3.3], for any Borel set $\Omega\subset\mathbb{R}^{d}$, we define its perimeter $\mathrm{Per}(\Omega)$ as

[TABLE]

We say that $\Omega$ is of finite perimeter if $\mathrm{Per}(\Omega)<\infty$. It was shown [18, 19, 20] that if $\Omega\subset\mathbb{R}^{d}$ is an open set of finite Lebesgue measure and of finite perimeter, then

[TABLE]

where

[TABLE]

is the transition density of the Brownian motion $\mathbf{B}=(B_{t})_{t\geq 0}$ in $\mathbb{R}^{d}$. For a Lévy process $\mathbf{X}$ with Lévy measure $\nu$, we define the perimeter $\mathrm{Per}_{\mathbf{X}}(\Omega)$ with respect to $\mathbf{X}$ as

[TABLE]

In particular, to the isotropic (rotationally invariant) $\alpha$-stable process $\mathbf{S}^{(\alpha)}=(S^{(\alpha)}_{t})_{t\geq 0}$, $0<\alpha<1$, one associates the $\alpha$-perimeter which is defined as

[TABLE]

where

[TABLE]

It is known, cf. [8, Lemma 1] (see also [13] for the perimeter for the isotropic stable processes), that if $\mathbf{X}$ is of bounded variation and $\Omega\subset\mathbb{R}^{d}$ is an open set of finite Lebesgue measure and of finite perimeter, then $\mathrm{Per}_{\mathbf{X}}(\Omega)$ is also finite.

Let $G\subset\mathbb{R}^{d}$ be an open set and $f:G\rightarrow\mathbb{R}$ be integrable. The total variation of $f$ in $G$ is

[TABLE]

The directional derivative of $f$ in $G$ in the direction of $u\in\mathbb{S}^{d-1}$ is

[TABLE]

We will use $V_{u}(\Omega)$ to denote $V_{u}(1_{\Omega},\mathbb{R}^{d})$.

Now we recall the covariogram function $g_{\Omega}(y)$. Let

[TABLE]

It is easy to see that

[TABLE]

Moreover $g_{\Omega}\in C_{0}(\mathbb{R}^{d})$ (see [14, Proposition 2]).

Let

[TABLE]

By the Fubini-Tonelli theorem we have the following relationship between $\text{Per}_{\mathbf{X}}(\Omega)$ and $f_{\Omega}(\cdot)$:

[TABLE]

Here is a simple lemma about the behavior of $f_{\Omega}(y)$ as $|y|\rightarrow 0$.

Lemma 1**.**

If $\Omega\subset\mathbb{R}^{d}$ is an open set of finite Lebesgue measure $|\Omega|$, then

[TABLE]

Proof. Note that $1_{\Omega}(x)1_{\Omega^{c}}(x-y)\leq 1_{\Omega}(x)$ for all $y\in\mathbb{R}^{d}$ and $\int 1_{\Omega}(x)\mathrm{d}x=|\Omega|<\infty$. For each $x\in\Omega$ we have

[TABLE]

Hence the assertion follows from the dominated convergence theorem. $\Box$

The next lemma follows from [14, Proposition 5].

Lemma 2**.**

If $\Omega\subset\mathbb{R}^{d}$ is a Borel set of finite Lebesgue measure, then

[TABLE]

We end this section by recalling the concept of regularly varying functions. A function $f$ is said to be regularly varying of index $\alpha$ at infinity if for any $\lambda>0$,

[TABLE]

The family of regularly varying functions of index $\alpha$ at infinity is denoted by $\mathcal{R}_{\alpha}$.

3. Processes of bounded variation in $\mathbb{R}^{d}$

In this section we assume that $\mathbf{X}$ is a Lévy process of bounded variation in $\mathbb{R}^{d}$.

3.1. Heat content

In this subsection, we first extend [8, Theorem 3] when the drift $\gamma_{0}=0$. Suppose that $\mathbf{X}$ is a purely discontinuous Lévy process of bounded variation in $\mathbb{R}^{d}$, that is, $A=0$, $\gamma_{0}=0$ and $\int_{{\lVert x\rVert}\leq 1}{\lVert x\rVert}\nu(\mathrm{d}x)<\infty$.

The infinitesimal generator $\mathcal{L}$ on $C_{0}(\mathbb{R}^{d})$ of $\mathbf{X}$ is a linear operator defined by

[TABLE]

with domain $\mathrm{Dom}(\mathcal{L})$ consisting of all functions $f$ such that the right hand side of (3) exists. By [22, Theorem 31.5], we have $C_{0}^{2}(\mathbb{R}^{d})\subset\mathrm{Dom}(\mathcal{L})$. For a detailed discussion on infinitesimal generators of Lévy processes we refer the reader to [22, Section 31]. Since $\mathbf{X}$ is of bounded variation and $\gamma_{0}=0$, again by [22, Theorem 31.5] we have, for any $f\in C_{0}^{2}(\mathbb{R}^{d})$,

[TABLE]

The next lemma corresponds to [8, Lemma 2].

Lemma 3**.**

Suppose that $\Omega\subset\mathbb{R}^{d}$ is an open set of finite Lebesgue measure. If

[TABLE]

then $g_{\Omega}(x)\in{\rm Dom}(\mathcal{L})$ and

[TABLE]

Proof. Fix a cut-off function $\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$ such that $\phi\geq 0$, $\mathrm{supp}\phi\subset B(0,1)$ and $\|\phi\|_{1}=1$. Let $\phi_{\varepsilon}(x):=\frac{1}{\varepsilon^{d}}\phi(\frac{x}{\varepsilon})$. Let $g_{\Omega}^{\varepsilon}(x):=g_{\Omega}*\phi_{\varepsilon}(x)$. Since $g_{\Omega}$ is integrable, we get that $g_{\Omega}^{\varepsilon}$ is smooth and vanishes at infinity. Hence $f_{\Omega}^{\varepsilon}\in C_{0}^{\infty}(\mathbb{R}^{d})$. Since $C_{0}^{2}(\mathbb{R}^{d})\subset\text{Dom}(\mathcal{L})$, we get $g_{\Omega}^{\varepsilon}\in\text{Dom}(\mathcal{L})$. Note that it follows from Lemma 2 that

[TABLE]

Lemma 1 implies

[TABLE]

By (4),

[TABLE]

Note that from Lemma 2 we have

[TABLE]

Hence, by the dominated convergence theorem we infer that

[TABLE]

Since $\mathcal{L}$ is a closed operator, the assertion of the lemma is now established. $\Box$

The following result is similar in spirit to [8, Theorem 3]. The difference is that in the result below we do not assume ${\rm Per}(\Omega)<\infty$ but we assume that $\gamma_{0}=0$.

Theorem 4**.**

Let $\mathbf{X}$ be a Lévy process of bounded variation with $\gamma_{0}=0$. If $\Omega\subset\mathbb{R}^{d}$ is an open set of finite Lebesgue measure, then we have

[TABLE]

Proof. First we assume that

[TABLE]

In this case the proof is similar to that of [8, Theorem 3]. It follows from Lemma 3 and (2) that

[TABLE]

Now we deal with the case when

[TABLE]

Note that

[TABLE]

Let $\varepsilon>0$, and $\phi_{\varepsilon}\in C_{b}(\mathbb{R}^{d})$ be such that $1_{B(0,\varepsilon)^{c}}\leq\phi_{\varepsilon}\leq 1_{B(0,\varepsilon/2)^{c}}$. Then by [22, Corollary 8.9] we get

[TABLE]

Since $\varepsilon$ is arbitrary, we have

[TABLE]

$\Box$

3.2. Spectral heat content

In this subsection we study the asymptotic behavior of the spectral heat content for Lévy processes of bounded variation. The main result is the following theorem.

Theorem 5**.**

Let $\mathbf{X}$ be a Lévy process of bounded variation in $\mathbb{R}^{d}$. If $\Omega\subset\mathbb{R}^{d}$ is an open set of finite measure $|\Omega|$ and of finite perimeter $\mathrm{Per}(\Omega)$, then

[TABLE]

where $V_{u}(\Omega)$ is the directional derivative of $1_{\Omega}$ in the direction $u$ on the unit sphere in $\mathbb{R}^{d}$.

Proof.

Observe that

[TABLE]

where

[TABLE]

By [8, Theorem 3], it suffices to show that

[TABLE]

By the Ikeda-Watanabe formula [17], the joint distribution of $(\tau_{\Omega},X_{\tau_{\Omega}})$ restricted to $X_{\tau_{\Omega}-}\neq X_{\tau_{\Omega}}$ is equal to

[TABLE]

where $p^{\Omega}_{s}(x,\mathrm{d}u)$ is the transition kernel of the process $\mathbf{X}$ killed upon exiting $\Omega$. Hence

[TABLE]

where

[TABLE]

Notice that

[TABLE]

Thus

[TABLE]

By the dominated convergence theorem and the right continuity of $\mathbf{X}$,

[TABLE]

$\Box$

As a consequence of Theorem 4, one can prove the following result. Note that, unlike Theorem 5, we do not assume that $\mathrm{Per}(\Omega)<\infty$ in the result below.

Theorem 6**.**

Let $\mathbf{X}$ be a Lévy process of bounded variation with $\gamma_{0}=0$. If $\Omega\subset\mathbb{R}^{d}$ is an open set of finite Lebesgue measure, then we have

[TABLE]

and

[TABLE]

In particular if $\mathbb{P}^{x}(X_{\tau_{\Omega}}\in\partial\Omega)=0$ for almost every $x\in\Omega$ we have

[TABLE]

Proof. If $\int_{\Omega}\nu(\Omega^{c}-y)\mathrm{d}y<\infty$, the proof is the same as the proof of Theorem 5 using Theorem 4 instead of [8, Theorem 3]. The case of $\int_{\Omega}\nu(\Omega^{c}-y)\mathrm{d}y=\infty$ is a consequence of Theorem 4 and the fact that $H_{\Omega}(t)\geq Q_{\Omega}(t)$.

$\Box$

Combining the above result with [23, Theorem 1], we immediately get the following

Corollary 7**.**

Suppose that $\mathbf{X}$ is an isotropic Lévy process of bounded variation and has an infinite Lévy measure. If $\Omega$ is a Lipschitz domain of finite Lebesgue measure, then

[TABLE]

Combining Theorem 6 with [7], we get the following

Corollary 8**.**

Let $d=1$ and $\Omega$ be open. Assume that $\mathbf{X}$ has an infinite Lévy measure, is of bounded variation and $\gamma_{0}=0$, then

[TABLE]

Proof. We use here that $\{0\}$ is polar ([7, Theoreme 8]) and therefore $\partial\Omega$ is polar. $\Box$

If there exists a nonzero drift $\gamma_{0}$, then the asymptotic behavior of the heat content and the spectral heat content can be different. We illustrate this by the simple example below.

Example 1**.**

Let $\gamma\in\mathbb{R}\setminus\{0\}$. We consider $\Omega=(-1,0)\cup(0,1)$ and a deterministic process $X_{t}=\gamma t$. Then $H(t)=(|\gamma|t)\wedge 2$ and $|\Omega|-Q_{\Omega}(t)=(2|\gamma|t)\wedge 2$. That is

[TABLE]

4. Processes of unbounded variation in $\mathbb{R}$

In this section we study the asymptotic behavior of the spectral heat content for symmetric Lévy processes on the real line. We consider two different cases separately. In the first case, we assume that the characteristic exponent $\psi(\xi)$ of $\mathbf{X}$ is regularly varying of index $\alpha$ at infinity for some $\alpha\in(1,2]$. In the second case, we assume that $\mathbf{X}$ is a symmetric Lévy process whose the characteristic exponent is $\psi(\xi)=|\xi|$, that is, $\mathbf{X}$ is a Cauchy process.

For any $\epsilon>0$, let $\Omega_{\epsilon}:=\{x\in\Omega:\mathrm{dist}(\{x\},\partial\Omega)<\epsilon\}$.

Lemma 9**.**

For any $\varepsilon>0$, we have

[TABLE]

Proof. It follows from (1) that

[TABLE]

$\Box$

Every open set $\Omega$ in $\mathbb{R}$ can be written as the union of countably many disjoint open intervals: $\Omega=\cup_{i}(a_{i},b_{i})$. Let $\Omega=\cup_{i}(a_{i},b_{i})$ and $\partial^{ad}\Omega:=\{b_{j}:\text{ there exists }i\neq j\text{ such that }b_{j}=a_{i}\}$ be the subset of $\partial\Omega$ which consists of common boundary points of adjacent components of $\Omega$. Let

[TABLE]

be the augmented set of $\Omega$. Note that the distance between any two distinct components of $\tilde{\Omega}$ is always strictly positive.

Recall that

[TABLE]

For any Lévy process $\mathbf{X}$ and $t\geq 0$, we define $\overline{X}_{t}:=\sup_{s\in[0,t]}X_{s}$ and $\underline{X}_{t}:=\inf_{s\in[0,t]}X_{s}$.

Let $\psi^{*}(u)=\sup_{\xi\in[0,u]}\psi(\xi)$, $u\geq 0$, and let $\psi^{-1}(u)=\inf\{s\geq 0:\psi^{*}(s)\geq u\}$ be the generalized inverse of $\psi^{*}$.

4.1. $\psi\in\mathcal{R}_{\alpha}$, $\alpha\in(1,2]$

In this subsection we study the asymptotic behavior of the spectral heat content of general open sets of finite Lebesgue measure with respect to symmetric Lévy processes in $\mathbb{R}$. We assume that $\mathbf{X}$ is a symmetric Lévy process with the characteristic exponent $\psi$ and that there exists $\alpha\in(1,2]$ such that $\psi\in\mathcal{R}_{\alpha}$.

For $y\in\mathbb{R}$, let $T_{y}=\inf\{t>0:X_{t}=y\}$ be the first time the process $\mathbf{X}$ hits $y$ and $T^{(\alpha)}_{y}$ the first time the symmetric $\alpha$-stable process $\mathbf{S}^{(\alpha)}$ (with characteristic exponent $|\xi|^{\alpha}$) hits $y$.

Here is the main result of this section:

Theorem 10**.**

Suppose that $\mathbf{X}$ is a symmetric Lévy process with the characteristic exponent $\psi\in\mathcal{R}_{\alpha}$ for some $\alpha\in(1,2]$. Let $\Omega$ be an open set in $\mathbb{R}$ with $|\Omega|<\infty$. Let $A$ be the number of components of $\tilde{\Omega}$ and $B$ be number of points in $\partial^{ad}\Omega$. Then we have

[TABLE]

where $C_{1}=\int_{0}^{\infty}\mathbb{P}(T_{u}^{(\alpha)}\leq 1)\mathrm{d}u\leq\mathbb{E}[\overline{S^{(\alpha)}}_{1}]<\infty$.

In the case of isotropic $\alpha$-stable processes, we have $\psi^{-1}(1/t)=t^{-1/\alpha}$.

Remark 11**.**

Note that $A$ can be finite even if the number of components in $\Omega$ is infinite. Also if $\Omega$ has infinitely many components, either $A$ or $B$ must be infinite and therefore we have $\displaystyle\lim_{t\rightarrow 0}\psi^{-1}(1/t)\left(|\Omega|-Q_{\Omega}(t)\right)=\infty$.

Under the assumptions of this subsection, the process $\mathbf{X}$ has a transition density, and thus $H_{\Omega}(t)=H_{\tilde{\Omega}}(t)$. Hence it follows from [8, Theorem 2] that when $\Omega\subset\mathbb{R}$ has infinitely many components but $\tilde{\Omega}$ has only finitely many components, we have

[TABLE]

Remark 12**.**

In the case of Brownian motion, we have

[TABLE]

Hence in this case (6) becomes

[TABLE]

We now give some preliminary results to prepare for the proof of Theorem 10. It is easy to check that in this case $\mathbf{X}$ is of unbounded variation. Since $\mathbf{X}$ is symmetric, it follows from [16, Corollary 1] that

[TABLE]

Also it follows from [6, Theorem 1.5.3] that, for each $R>0$, there exists a constant $c=c(R)>0$ such that

[TABLE]

Since $\psi\in\mathcal{R}_{\alpha}$ for some $\alpha\in(1,2]$ and $\mathbf{X}$ is symmetric, we have

[TABLE]

It follows from [7, Theoreme 8] that $\{y\in\mathbb{R}:\mathbb{P}(T_{y}<\infty)>0\}=\mathbb{R}$.

Lemma 13**.**

Suppose that $\mathbf{X}$ is a symmetric Lévy process with the characteristic exponent $\psi\in\mathcal{R}_{\alpha}$ for some $\alpha\in(1,2]$. There exists $\varepsilon_{1}>0$ such that for any $\varepsilon\leq\varepsilon_{1}$,

[TABLE]

Proof. Define a process $\mathbf{Y}^{(t)}=(Y^{(t)}_{s})_{s\geq 0}$ by $Y^{(t)}_{s}=\psi^{-1}(1/t)X_{ts}$. The characteristic exponent of $\mathbf{Y}^{(t)}$ is $\psi^{(t)}(\xi)=t\psi(\psi^{-1}(1/t)\xi)$. Note that

[TABLE]

Observe that by the change of variables $u=\psi^{-1}(1/t)y$,

[TABLE]

Fix $0<\delta<\alpha-1$. Since $\psi\in\mathcal{R}_{\alpha}$, it follows from [6, Theorem 1.5.6] that there exists $x_{0}>0$ such that

[TABLE]

for all $\psi^{-1}(1/t)\geq x_{0}$ and $1\leq u\leq\frac{\psi^{-1}(1/t)}{x_{0}}$. Hence by [5, Theorem II.19.(iii)] and the dominated convergence theorem,

[TABLE]

Let $\varepsilon_{1}:=\frac{1}{x_{0}}$. It follows from (8) that there exists $c=c(x_{0})>0$ such that

[TABLE]

Take $M>0$. It follows from the dominated convergence theorem and (10) that

[TABLE]

Suppose $\varepsilon\leq\varepsilon_{1}$ and $M\leq u\leq\varepsilon\psi^{-1}(1/t)$. Then from (1), (7), (9) and (11) we get that

[TABLE]

Hence we have

[TABLE]

for all $\varepsilon\leq\varepsilon_{1}$, $M<\varepsilon\psi^{-1}(1/t)$, and $\psi^{-1}(1/t)\geq x_{0}$. By letting $t\rightarrow 0$ and then letting $M\rightarrow\infty$ in (12) and (13), hen the conclusion of the lemma. $\Box$

Lemma 14**.**

Suppose that $\mathbf{X}$ is a symmetric Lévy process with the characteristic exponent $\psi\in\mathcal{R}_{\alpha}$ for some $\alpha\in(1,2]$. There exists $\varepsilon_{2}>0$ such that for all $\varepsilon\leq\varepsilon_{2}$,

[TABLE]

Proof. Let $\mathbf{Y}^{(t)}$ be the process defined in the proof of Lemma 13. Recall from the proof of Lemma 13 that $\psi^{(t)}$ is the characteristic exponent of $\mathbf{Y}^{(t)}$ and

[TABLE]

Moreover,

[TABLE]

Using (1), (14), and [15, Theorem VI.5.5], we can get that $\overline{Y^{(t)}}_{1}\xrightarrow{D}\overline{S^{(\alpha)}}_{1}$ (since $x\mapsto\sup_{t\in[0,1]}x(t)$ is a continuous functional on the Skorohod space). The rest of the proof is identical to the proof of Lemma 13. $\Box$

Lemma 15**.**

Suppose that $\mathbf{X}$ is a symmetric Lévy process with the characteristic exponent $\psi\in\mathcal{R}_{\alpha}$ for some $\alpha\in(1,2]$. Let $\Omega=\cup_{i}(a_{i},b_{i})$ with $|\Omega|=\sum_{i}(b_{i}-a_{i})<\infty$. Suppose that $a_{i}\notin\partial^{ad}\Omega$ and $\varepsilon<\frac{1}{2}((b_{i}-a_{i})\wedge\varepsilon_{2})$. Then we have

[TABLE]

Similarly, if $b_{i}\notin\partial^{ad}\Omega$ and $\varepsilon<\frac{1}{2}((b_{i}-a_{i})\wedge\varepsilon_{2})$, then

[TABLE]

Proof. By the symmetry of X it is enough to prove the first limit. Suppose that $a_{i}\notin\partial^{ad}\Omega$, $\varepsilon<\frac{b_{i}-a_{i}}{2}$ and $x\in(a_{i},a_{i}+\varepsilon)$. Note that, under $\mathbb{P}_{x}$, the event $\{\tau_{\Omega}>t\}$ can be written as

[TABLE]

Note that the first event of the display above is disjoint with the union of the last two events. Hence we have

[TABLE]

This implies that

[TABLE]

Let $b:=\sup\{x\in\Omega:x<a_{i}\}$. Since $a_{i}\notin\partial^{ad}\Omega$, we have either $\{x\in\Omega:x<a_{i}\}=\emptyset$ thus $b=-\infty$ or $b<a_{i}$. Hence we have either

[TABLE]

or

[TABLE]

We will deal with the second case since the first case is similar and much easier. It follows from (16) that (15) can be written as

[TABLE]

Hence

[TABLE]

Note that by the symmetry of $\mathbf{X}$ we have

[TABLE]

Hence from Lemma 14,

[TABLE]

By (1) we have

[TABLE]

Since $\psi^{-1}\in\mathcal{R}_{1/\alpha}$ (see [6, Theorems 1.5.3 and 1.5.12]), we have

[TABLE]

Since $a_{i}-b>0$, by the symmetry of $\mathbf{X}$ and the same argument as above we get

[TABLE]

The proof is now complete. $\Box$

Lemma 16**.**

Suppose that $\mathbf{X}$ is a symmetric Lévy process with the characteristic exponent $\psi\in\mathcal{R}_{\alpha}$ for some $\alpha\in(1,2]$. Let $\Omega=\cup_{i}(a_{i},b_{i})$ with $|\Omega|=\sum_{i}(b_{i}-a_{i})<\infty$. If $a_{i}\in\partial^{ad}\Omega$ and $\varepsilon<\frac{1}{2}((b_{i}-a_{i})\wedge\varepsilon_{1})$, then

[TABLE]

Similarly, if $b_{i}\in\partial^{ad}\Omega$ and $\varepsilon<\frac{1}{2}((b_{i}-a_{i})\wedge\varepsilon_{1})$, then

[TABLE]

Proof. Suppose that $a_{i}\in\partial^{ad}\Omega$, $\varepsilon<\frac{b_{i}-a_{i}}{2}$ and $x\in(a_{i},a_{i}+\varepsilon)$. Let $(a_{j},b_{j})$ with $b_{j}=a_{i}$ be the component of $\Omega$ which is adjacent to $(a_{i},b_{i})$. Then we have under $\mathbb{P}_{x}$,

[TABLE]

It follows from an argument similar to that in the proof of Lemma 15 we have

[TABLE]

Hence we have

[TABLE]

and

[TABLE]

Now using Lemma 13 we obtain the claim. $\Box$

Now we state a result handling the case when $\Omega$ has infinitely many components.

Lemma 17**.**

Suppose that $\mathbf{X}$ is a symmetric Lévy process with the characteristic exponent $\psi\in\mathcal{R}_{\alpha}$ for some $\alpha\in(1,2]$. If $\Omega\subset\mathbb{R}$ is of finite Lebesgue measure and has infinitely many components, then

[TABLE]

Proof. If $\Omega$ has infinitely many components, either $A$, the number of components in $\tilde{\Omega}$, or $B$, the number of points in $\partial^{ad}\Omega$, is infinite. Suppose that $A=\infty$. Let $\Omega=\cup_{i=1}^{\infty}(a_{i},b_{i})$. Then there must be infinitely many $i$ such that $a_{i}\notin\partial^{ad}\Omega$ or $b_{i}\notin\partial^{ad}\Omega$. Let $\mathcal{I}=\{i:a_{i}\notin\partial^{ad}\Omega\text{ or }b_{i}\notin\partial^{ad}\Omega\}$. Given $N$, take $\varepsilon=\varepsilon(N)$ small so that there are at least $N$ many $i$’s with $\varepsilon<\frac{1}{2}((b_{i}-a_{i})\wedge\varepsilon_{1})$. Then it follows from Lemma 15 we have

[TABLE]

Now the assertion follows by letting $N\rightarrow\infty$.

The case when $B=\infty$ can be proved in a similar way using Lemma 16. $\Box$

Now we are ready to prove Theorem 10.

Proof of Theorem 10 If $\Omega$ has infinitely many components, the result follows from Lemma 17. Now assume that $\Omega$ has finitely many components. Write $\Omega=\cup_{i=1}^{N}(a_{i},b_{i})$ and let $\varepsilon=\frac{1}{2}\min_{1\leq i\leq N}\left((b_{i}-a_{i})\wedge\varepsilon_{1}\wedge\varepsilon_{2}\right)$, where $\varepsilon_{1}$ and $\varepsilon_{2}$ are constants in Lemmas 13 and 14, respectively. Let $\mathcal{B}$ be the set of points which are the common end point of two adjacent components of $\Omega$ and $\mathcal{A}=\cup_{i=1}^{N}\{a_{i},b_{i}\}\setminus\mathcal{B}$. Then $|\mathcal{A}|=2A$, $|\mathcal{B}|=B$, and $A+B=N$. It follows from Lemmas 9, 15 and 16 that

[TABLE]

$\Box$

4.2. Cauchy process

First we give some preliminary results to prepare for the proof of Theorem 21. In this subsection we assume that $\mathbf{X}$ is a Cauchy process, that is, a symmetric 1-stable Lévy process, in $\mathbb{R}$ with the characteristic exponent $\psi(\xi)=|\xi|$.

Lemma 18**.**

Suppose that $\mathbf{X}$ is a Cauchy process. Let $\Omega=\cup_{i}(a_{i},b_{i})$ with $|\Omega|=\sum_{i}(b_{i}-a_{i})<\infty$. If $a_{i}\notin\partial^{ad}\Omega$ and $\varepsilon<\frac{b_{i}-a_{i}}{2}$, then

[TABLE]

Similarly, if $b_{i}\notin\partial^{ad}\Omega$ and $\varepsilon<\frac{b_{i}-a_{i}}{2}$, then

[TABLE]

Proof. The proof is almost identical to the proof of Lemma 15 using [3, Proposition 4.3 (i)] instead of Lemma 14, so we omit the details. $\Box$

Now we address the issue when $\Omega$ has adjacent components. Recall the definition of augmented set $\tilde{\Omega}$ in (5). It is well known that, when $0<\alpha\leq 1$, a single point is polar for the process hence $T_{x}=\inf\{s:X_{s}=x\}$ is almost surely infinite. Hence we have the following result.

Lemma 19**.**

If $\mathbf{X}$ is a Cauchy process, then $Q_{\Omega}(t)=Q_{\tilde{\Omega}}(t)$.

Proof. By ([7, Theoreme 8]) $\{0\}$ is polar and therefore $\partial\Omega$ is polar as well. Hence

[TABLE]

almost surely. This implies the claim. $\Box$

Lemma 20**.**

Suppose that $\mathbf{X}$ is a Cauchy process. If $\Omega$ is of finite Lebesgue measure and $\tilde{\Omega}=\Omega\cup\partial^{ad}\Omega$ has infinitely many components, then

[TABLE]

Proof. The proof is very similar to the proof of Lemma 17 using Lemma 18. $\Box$

Theorem 21**.**

Let $\mathbf{X}$ be a Cauchy process. Let $\Omega=\cup_{i}(a_{i},b_{i})$ be an open set in $\mathbb{R}$ with $|\Omega|=\sum_{i}(b_{i}-a_{i})<\infty$. Let $A$ be the number of components of $\tilde{\Omega}$. Then we have

[TABLE]

Proof. The proof is very similar to the proof of Theorem 10 using Lemmas 9, 18 and 19, 20 and we omit the details. $\Box$

5. Perturbation Results

In this section, we assume that $\mathbf{Y}$ is a Lévy process in $\mathbb{R}^{d}$ with the Lévy triplet $(A,\gamma,\nu^{Y})$ such that

[TABLE]

where $\lim_{t\to 0}\frac{t}{f(t)}=0$. Throughout this section, the superscript $\mathbf{Y}$ always means quantities corresponding to the process $\mathbf{Y}$.

Now we assume that $\mathbf{X}$ is a Lévy process in $\mathbb{R}^{d}$ with the Lévy triplet $(A,\gamma,\nu)$ such that the signed measure $\sigma(\mathrm{d}x):=\nu(\mathrm{d}x)-\nu^{Y}(\mathrm{d}x)$ has finite total variation $m$.

Here is the main result of this section.

Theorem 22**.**

We have

[TABLE]

In order to prove Theorem 22 we need two lemmas.

Lemma 23**.**

If $\sigma(\mathrm{d}x)$ is a nonnegative measure, then for any $t>0$ we have

[TABLE]

Proof.

Write $X_{t}=Y_{t}+V_{t}$, where $\mathbf{V}=(V_{t})_{t\geq 0}$ is a compound Poisson process independent of $\mathbf{Y}$. Let $T=\inf\{s\geq 0:V_{s}\neq 0\}$. Since $\tau_{\Omega}^{\mathbf{Y}}$ and $T$ are independent, we have

[TABLE]

This establishes the claim of the lemma. $\Box$

Lemma 24**.**

If $\sigma(\mathrm{d}x)$ is a nonnegative measure, then for any $t>0$ we have

[TABLE]

Proof.

As in the proof of Lemma 23, we write $X_{t}=Y_{t}+V_{t}$, where $\mathbf{V}$ is a compound Poisson process independent of $\mathbf{Y}$. Then by independence of $\mathbf{Y}$ and $\mathbf{V}$, we have

[TABLE]

where we used the fact that $\{\tau_{\Omega}^{\mathbf{Y}}>t,T>t\}\subset\{\tau_{\Omega}>t\}$ in the last inequality. Hence we have $Q^{\mathbf{Y}}_{\Omega}(t)\leq e^{mt}Q_{\Omega}(t)$ and this immediately implies $|\Omega|-Q^{\mathbf{Y}}_{\Omega}(t)\geq|\Omega|-e^{mt}Q_{\Omega}(t)$. $\Box$

Now we are ready to prove Theorem 22.

Proof of Theorem 22. By assumption the signed measure $\sigma(\mathrm{d}x)$ has finite total variation. Let $\sigma(\mathrm{d}x)=\sigma^{+}(\mathrm{d}x)-\sigma^{-}(\mathrm{d}x)$ be the Hahn-Jordan decomposition (see [11, Theorem 3.3 and 3.4]) of $\sigma(\mathrm{d}x)$ such that $\mathcal{P}\cup\mathcal{N}=\mathbb{R}^{d}$, $\mathcal{P}\cap\mathcal{N}=\emptyset$, and $\sigma^{+}(\mathcal{N})=\sigma^{-}(\mathcal{P})=0$. Let $\mathbf{Z}$ be a Lévy process with Lévy density $\nu^{\mathbf{Z}}(\mathrm{d}x)=\nu(\mathrm{d}x)1_{\mathcal{N}}(x)+\nu^{\mathbf{Y}}(\mathrm{d}x)1_{\mathcal{P}}(x)$. Note that $\sigma_{+}(\mathrm{d}x):=\nu(\mathrm{d}x)-\nu^{\mathbf{Z}}(\mathrm{d}x)$ is a nonnegative measure on $\mathbb{R}^{d}$ and

[TABLE]

Hence from Lemmas 23 and 24 we have

[TABLE]

By interchanging the role of $\mathbf{X}$ and $\mathbf{Y}$ we also have

[TABLE]

where $\sigma_{-}(\mathrm{d}x):=\nu^{\mathbf{Y}}(\mathrm{d}x)-\nu^{\mathbf{Z}}(\mathrm{d}x)$ and $m_{2}:=\int_{\mathbb{R}^{d}}\sigma_{-}(\mathrm{d}x)=\int_{\mathcal{N}}\sigma^{-}(\mathrm{d}x)<\infty.$

Hence it follows from (5) and (18) we have

[TABLE]

and

[TABLE]

Since $\lim_{t\rightarrow 0}\frac{e^{mt}-1}{f(t)}=0$, we have

[TABLE]

and

[TABLE]

The proof is now complete. $\Box$

6. Examples

In this section we examine concrete examples of the asymptotic behavior of the spectral heat content for various Lévy processes.

Symmetric stable processes in $\mathbb{R}$ and their perturbations.

Recall that the Lévy measure of the symmetric $\alpha$-stable process in $\mathbb{R}$ is given by $\nu^{\mathbf{S}^{(\alpha)}}(\mathrm{d}x)=\frac{c(1,\alpha)}{|x|^{1+\alpha}}\mathrm{d}x$.

Now we assume that $\mathbf{X}$ is a Lévy process in $\mathbb{R}$ with Lévy triplet $(0,0,\nu)$ such that the signed measure $\sigma(\mathrm{d}x)=\nu(\mathrm{d}x)-\nu^{\mathbf{S}^{(\alpha)}}(\mathrm{d}x)$ has finite total variation $m$. Let

[TABLE]

Note that, when $0<\alpha<1$, the process X is of bounded variation. As a consequence of Corollary 8, Theorems 10, 21, and 22, we immediately get the following.

Proposition 25**.**

Suppose the assumptions in the paragraph above hold. Let $\Omega=\cup_{i}(a_{i},b_{i})$ be an open set in $\mathbb{R}$ with $|\Omega|=\sum_{i}(b_{i}-a_{i})<\infty$. Let $A$ be the number of components of $\tilde{\Omega}$ and $B$ be number of points in $\partial^{ad}\Omega$. Then we have

[TABLE]

Remark 26**.**

Proposition 25 is a natural generalization of the main result in [3]. We remark here that the set $\Omega\subset\mathbb{R}$ is an arbitrary open in $\mathbb{R}$ of finite Lebesgue measure. The class of process we are dealing with here is much larger than the class of symmetric stable processes.

Fractional perimeter for symmetric stable processes in $\mathbb{R}$.

1. (i)

If $\Omega\subset\mathbb{R}$ has finitely many components, then $\Omega$ has a finite perimeter, which is equivalent to $f_{\Omega}(y)$ is Lipschitz (see [14]), we have $\int_{\mathbb{R}^{d}}\frac{f_{\Omega}(y)}{|y|^{d+\alpha}}\mathrm{d}y<\infty$. Hence Theorem 4 recovers [8, Theorem 3]. 2. (ii)

Now we give an example of an open set $\Omega\subset\mathbb{R}$ with $|\Omega|<\infty$ such that $\mathrm{Per}_{\mathbf{S}^{(\alpha)}}(\Omega)=\infty$ for all $0<\alpha<1$. Let $\{d_{n}\}_{n\in\mathbb{N}}\subset(0,1)$ be a strictly decreasing sequence such that $\sum_{n=1}^{\infty}d_{n}<\infty$. We consider an open set $\Omega:=\cup_{n=1}^{\infty}(n,n+d_{n}).$ Define for each $y\in(0,1)$ a number $n(y)=\sup\{k:d_{k}\geq y\}.$ We have $g_{\Omega}(y)=(d_{1}+d_{2}+\cdots+d_{n(y)})-n(y)y,$ and

[TABLE]

Now we fix $b>1$ and consider $d_{n}=\frac{1}{n(1+\ln n)^{b}}$. Using the definition of $n(y)$ it is easy to check that

[TABLE]

Thus for any $0<\alpha<1$ we have

[TABLE] 3. (iii)

Finally we state a simple criteria that guarantees $\mathrm{Per}_{\mathbf{S}^{(\alpha)}}(\Omega)<\infty$.

Lemma 27**.**

Let $\alpha\in(0,1)$. Suppose that $\Omega=\cup_{i=1}^{\infty}\Omega_{i}\subset\mathbb{R}$, where $\Omega_{i}$ are open connected and disjoint. If $\sum_{i=1}^{\infty}|\Omega_{i}|^{1-\alpha}<\infty$, then

[TABLE]

Proof. Note that for $x\in\Omega$ we have

[TABLE]

Hence

[TABLE]

$\Box$

Let consider $\Omega=\cup_{n=1}^{\infty}(n,n+\frac{1}{n^{b}})$ with $b>1$. By the above lemma $\mathrm{Per}_{\mathbf{S}^{(\alpha)}}\Omega<\infty$ if $b>1/(1-\alpha)$. Hence if $b>1/(1-\alpha)$, then we have

[TABLE]

On the other hand, using an argument similar to that leading to (19), we get $\mathrm{Per}_{\mathbf{S}^{(\alpha)}}\Omega=\infty$, for $b\leq 1/(1-\alpha)$. Hence we conclude that

[TABLE]

Isotropic $\alpha$-stable processes in $\mathbb{R}^{d}$, $\alpha\in(0,1)$ and $d\geq 2$.

Consider an isotropic $\alpha$- stable process $\mathbf{S}^{(\alpha)}$ with $0<\alpha<1$ on $\mathbb{R}^{d}$, $d\geq 2$. Suppose that the open set $\Omega$ satisfies the following volume density condition (see [30, Equation (1.1)])

[TABLE]

for some constant $c>0$ and $|\partial\Omega|=0$. Then it follows from [30, Theorem 1] that $\mathbb{P}_{x}(\tau_{\Omega}^{\mathbf{S}^{(\alpha)}}\in\partial\Omega)=0$ for all $x\in\Omega$. Hence by Theorem 6 we have

[TABLE]

Note that any Lipschitz open sets satisfy volume density condition.

4. Relativistic stable processes.

Suppose that $\mathbf{X}^{m}$ is a relativistic $\alpha$-stable process with mass $m>0$ whose the characteristic exponent is

[TABLE]

Let $\nu^{m}(x)$ be the Lévy density of $\mathbf{X}^{m}$. It is well-known that $0<\nu^{m}(x)\leq\nu^{\mathbf{S}^{(\alpha)}}(x)$ and

[TABLE]

Hence if $\alpha\geq 1$ and $d=1$, it follows from Proposition 25 that

[TABLE]

On the other hand when $0<\alpha<1$ and $\Omega$ is a Lipschitz open set in $\mathbb{R}^{d}$, $d\geq 2$ or an arbitrary open set in $\mathbb{R}$ it follows from Corollaries 7 and 8 that

[TABLE]

5. Truncated stable processes.

Let $\mathbf{X}^{T}$ be a truncated $\alpha$-stable process with Lévy triplet $(0,0,\nu^{\mathbf{X}^{T}})$, where

[TABLE]

By the same argument as in the case of relativistic stable processes, we get that when $\alpha\geq 1$ and $d=1$, we have

[TABLE]

When $0<\alpha<1$ and $\Omega$ is a Lipschitz open set in $\mathbb{R}^{d}$, $d\geq 2$ or an arbitrary open set in $\mathbb{R}$ it follows from Corollaries 7 and 8 that

[TABLE]

6. Logarithmic perturbations.

Let $\mathbf{X}^{L}$ be a Lévy process with Lévy triplet $(0,0,\nu^{\mathbf{X}^{L}})$, where

[TABLE]

where $\beta\in\mathbb{R}$. By [10, Proposition 2] we have that $\psi^{L}\in\mathcal{R}_{\alpha}$ and $\psi^{L}(s)\sim s^{\alpha}\ln^{\beta}(s)$, where $f(s)\sim g(s)$ means $\displaystyle\lim_{s\rightarrow\infty}\frac{f(s)}{g(s)}=1$. This and [6, Proposition 1.5.15] imply $(\psi^{L})^{-1}(s)\sim s^{1/\alpha}\ln^{-\beta/\alpha}(s)$. Hence we get by Theorem 10, for $\alpha>1$

[TABLE]

When $\alpha<1$ or $\alpha=1$ and $\beta<-1$ the process $\textbf{X}^{L}$ is of bounded variation, therefore one can apply Theorem 6 and Corollaries 8 and 7 in this case.