Barriers, exit time and survival probability for unimodal Lévy processes

Abstract

We give superharmonic functions and derive sharp bounds for the expected exit time and probability of survival for isotropic unimodal Lévy processes in smooth domains.

1 Introduction

A function is called barrier for an open set if it is superharmonic inside and vanishes outside, near a part of the boundary of the set. Barriers are important for studying boundary behavior of solutions to the Dirichlet problem [2, 4]. From a general perspective, understanding boundary asymptotics of superharmonic functions gives detailed information on the behavior of the underlying Markov process at the boundary. The information is obtained by using maximum principle, super-mean value property and Doob’s conditioning. Calculation of barriers is extremely delicate for open sets with Lipschitz regularity, even for the Laplacian and cones in ${\mathbb {R}^{d}}$, see, e.g., [3], [14, Section 3]. The situation is somewhat easier for smooth open sets. For instance, the Laplacian in a half-space has barriers which are linear functions, correspondingly for smooth sets approximately linear barriers exist. Similar results, with non-linear boundary decay, are known for the fractional Laplacian and generators of convolution semigroups corresponding to complete subordinate Brownian motions with weak scaling (see [34, 38] and Sect. 7 for discussion and references). Recall that for a sub-Markovian semigroup $(P_t,t\geqslant 0)$ we have ${\mathcal {A}}f(x)=\lim _{t\rightarrow 0^+} [P_tf(x)-f(x)]/t\leqslant 0$ if $f$ is bounded, the limit exists and $f(x)=\max f\geqslant 0$. Accordingly, we say that operator ${\mathcal {A}}$ on $C^\infty _c({\mathbb {R}^{d}})$ satisfies the positive maximum principle if for every $\varphi \in C^\infty _c({\mathbb {R}^{d}})$, $\varphi (x)=\sup _{y\in {\mathbb {R}^{d}}} \varphi (y)\geqslant 0$ implies ${\mathcal {A}}\varphi (x)\leqslant 0$. The most general operators which have this property are of the form

$$\begin{aligned} {\mathcal {A}}\varphi (x)&= \sum _{i,j=1}^d a_{ij}(x)D_{x_i}D_{x_j} \varphi (x) +b(x) \nabla \varphi (x) +q(x)\varphi (x)\\&+\int \limits _{{\mathbb {R}^{d}}}\left( \varphi (x+y)-\varphi (x)- y\nabla \varphi (x)\;\mathbf{1}_{|y|<1} \right) \, \nu (x,dy). \end{aligned}$$

Here for every $x\in {\mathbb {R}^{d}}$, $a(x)=(a_{ij}(x))_{i,j=1}^n$ is a real nonnegative definite symmetric matrix, vector $b(x)=(b_i(x))_{i=1}^d$ has real coordinates, $q(x)\leqslant 0$, and $\nu (x,\cdot )$ is a Lévy measure. The description is due to Courrège, see, e.g., [28, Proposition 2.10]. For translation invariant (convolution) operators of this type, $a$, $b$, $q$, and $\nu$ are independent of $x$. If we further assume rotation invariance and conservativeness (${\mathcal {A}}1=0$), then

$$\begin{aligned} {\mathcal {A}}\varphi (x)&= \sigma \Delta \varphi (x) +\lim _{\varepsilon \rightarrow 0^+}\int \limits _{|y|>\varepsilon }\left[ \varphi (x+y)-\varphi (x)\right] \nu (dy), \end{aligned}$$

(1.1)

where $\sigma \geqslant 0$ and $\nu$ is isotropic. (1.1) gives the general setting of our paper; we shall also consider the corresponding isotropic Lévy processes $X$.

It is in general difficult to determine barriers for non-local Markov generators, even in the setting of (1.1) and for smooth open sets. In fact the wide range of Lévy measures $\nu$ results in a comparable variety of boundary asymptotics of superharmonic functions, not fully codified by the existing calculus. The situation might even seem hopeless but it is not. For instance, the expected exit time $x\mapsto \mathbb {E}_x \tau _D$ of $X$ from open bounded set $D\subset {\mathbb {R}^{d}}$ is a barrier for $D$. We shall effectively estimate this function for smooth open sets $D$ and unimodal Lévy processes $X$ by giving barriers for the ball of arbitrary radius. To this end we use the renewal function $V$ of the ladder-height process of one-dimensional projections of $X$: the barriers are defined as compositions of $V$ with the distance to the complement of the ball. This and a similar definition of functions subharmonic in the complement of the ball yield sharp estimates for the expected exit time for open sets $D\subset {\mathbb {R}^{d}}$ which are of class $C^{1,1}$. We also obtain sharp estimates for the probability of $X$ surviving in $D$ longer than given time $t>0$, even for some unbounded $D$ and rather general unimodal Lévy processes.

Thus, $V$ allows for calculations accurate enough to exhibit specific super- and subharmonic functions for the considered processes. The idea of using $V$ in this context comes from Kim et al. [34] (see Introduction and p. 931 ibid.) and has already proved very fruitful for complete subordinate Brownian motions.

When verifying superharmonicity, we calculate a version of the infinitesimal generator on the composition of $V$ with the distance to the complement of the ball. In view of the curvature of the sphere, the calculation requires good control of $V'$. We carry out calculations assuming that $V'$ satisfies a Harnack-type condition $(\mathbf{H})$, described in (3.7) below. When using $(\mathbf{H})$ we only need to estimate certain weighted integrals of $V'$ (given, e.g., by Lemma 3.7), rather than individual values of $V'$. The condition $(\mathbf{H})$ holds, e.g., for special subordinate Brownian motions, a class of processes wider than the complete subordinate Brownian motions. We should note that $V$ is defined implicitly but in the considered isotropic setting it enjoys simple sharp estimates in terms of more elementary functions: the Lévy–Khintchine exponent $\psi$ of $X$ and the following Pruitt’s function $h$ [41] (see (2.5) below for details),

$$\begin{aligned} h(r)&=\frac{\sigma ^2 d}{r^2}+\int \limits _{{\mathbb {R}^{d}}}\left( \frac{|z|^2}{r^2}\wedge 1\right) \nu (dz),\quad r>0. \end{aligned}$$

(1.2)

Namely, it follows from Proposition 2.4 and (3.1) that for unimodal Lévy processes with unbounded $\psi$ we have

$$\begin{aligned} h(r)\approx \psi (1/r)\approx 1/V(r)^{2},\quad r>0. \end{aligned}$$

(1.3)

On the other hand, the control of $V'$ is hard. For instance continuity and monotonicity of $V'$, although common, are open to conjectures (we actually know that $V'$ may fail to be monotone for some unimodal Lévy processes, see Remark 8). For complete subordinate Brownian motions good control results from the fact that $V'$ is completely monotone. This sheds light on the results obtained by Chen et al. (cf. [17, 36] and Sect. 7.2 below). Our approach allows to lift this structure requirement that $X$ is a subordinate Brownian motion, thanks to new ideas employing unimodality, scaling and (approximating) Dynkin’s operator.

The basic object of interest in our study is $\mathbb {E}^x\tau _{B_r}$, the expected exit time from the ball $B_r$ centered at the origin and with radius $r>0$, for arbitrary starting point $x\in {\mathbb {R}^{d}}$ of $X$ (for detailed definitions see Sect. 2). When $x=0$, the classical result of Pruitt [41] (see p. 954, Theorem 1 and (3.2) ibid.) provides in our setting constants $c=c(d)$ and $C=C(d)$ such that

$$\begin{aligned} \frac{c}{h(r)}\leqslant \mathbb {E}^0 \tau _{B_r}\leqslant \frac{C}{h(r)}, \qquad r>0. \end{aligned}$$

(1.4)

Pruitt’s estimate may be called sharp, meaning that the ratio of its extreme sides is bounded. One of our main contributions is the following inequality,

$$\begin{aligned} \frac{{c^*}}{\sqrt{h(r)h(r-|x|)} } \leqslant \mathbb {E}^x\tau _{B_r}\leqslant \frac{C}{\sqrt{h(r)h(r-|x|)}},\quad x\in B(0,r), \end{aligned}$$

(1.5)

where $c^*=c^*(r,d, X)>0$ is non-increasing in $r$ and $C=C(d)$. The estimate holds for unimodal Lévy processes under condition $(\mathbf{H})$ on $V'$. The estimate is sharp up to the boundary of the ball. As we note in Lemma 2.3, the upper bound in (1.5) easily follows from the one-dimensional case (2.18), cf. [26]. The lower bound is much more delicate. To the best of our knowledge the lower bound was only known for complete subordinate Brownian motions satisfying certain scaling conditions (see Theorem 1.2 and Proposition 2.7 in [30]). Our results cover in a uniform way isotropic stable process, relativistic stable process, sums of two independent isotropic stable processes (also with Gaussian component), geometric stable processes, variance gamma processes, conjugate to geometric stable processes [44] and much more which could not be treated by previous methods. The fact that $c$ in (1.5) depends on $r$ is a drawback if one needs to consider large $r$. In many situations, however, we may actually choose $c$ independent of $r$. For example if $X$ is a special subordinate Brownian motion, then we have $c=c(d)$, which follows by combining Theorem 4.1 with Lemma 7.5 below. We conjecture that in the case of isotropic Lévy processes, one can always choose $c$ depending only on $d$. This is certainly true in the one-dimensional case, see (2.18). For $d\geqslant 2$ the conjecture is strongly supported by comparison of (1.4) and (1.5).

We test super- and subharmonicity by means of Dynkin’s generator of $X$ in a way suggested by [13]. We also rely on our recent bounds for the semigroups of weakly scaling unimodal Lévy processes on the whole of ${\mathbb {R}^{d}}$ [10], and results of Grzywny [24]. As we indicated above, delicate properties of $V$, indeed of $V'$, are used to prove (1.5) by way of calculating Dynkin’s operator on functions defined with the help of $V$. Fortunately, the resulting asymptotics is directly expressed by $V$, rather than by $V'$, and may also be described by means of the Lévy–Khintchine exponent $\psi$ or $h$, which we indeed do in (1.5) (estimates expressed in terms of $h$ may be considered the most explicit, because $h$ is given by a direct integration without cancellations).

On a general level our development rests on estimates for Dynkin-type generators acting on smooth test functions (Sect. 2) and compositions of $V$ (Sect. 3). This explains our restriction to $C^{1,1}$ open sets: we approximate them by translations and rotations of the half-space $\mathbb {H}=\{x\in {\mathbb {R}^{d}}: x_1>0\}$, and $V(x_1)$ is harmonic for $X$ on $\mathbb {H}$. Noteworthy, the so-called boundary Harnack principle (BHP) for harmonic functions of $X$ is negligible in our development; it is superseded in estimates by the ubiquitous function $V$. Barriers resulting from $V$ provide access to asymptotics of the expected exit time, survival probability, Green function, harmonic measure, distribution of the exit time and the heat kernel. In fact, our estimates imply explicit decay rate for nonnegative harmonic functions near the boundary of $C^{1,1}$ open sets, see Proposition 7.6. Furthermore, in [8] we give applications to heat kernels for the corresponding Dirichlet problem in $C^{1,1}$ open sets. We also expect applications to Hardy-type inequalities, cf. [2].

It would be of considerable interest to further extend our estimates to Markov processes with isotropic Lévy kernels $dy\mapsto \nu (x,dy)$ or to isotropic Lévy processes with the Lévy measure approximately unimodal in the sense of (4.3). Partial results in this direction are given in Corollary 4.3. We like to note that the case of Lipschitz open sets apparently requires approach based on BHP and is bound to produce less explicit estimates. We refer the reader to [3, 9] for more information and bibliography on this subject. In this connection we like to note that BHP fails for non-convex open sets for the so-called truncated stable Lévy processes [32].

The rest of the paper is composed as follows. In Sect. 2 we estimate tails of $X_t$ and $X_{\tau _D}$ by means of $\mathbb {E}^x\tau _D$, $V$ or $h$. In Lemma 3.8 and 3.9 of Sect. 3 we give mildly super- and subharmonic functions for the ball and the complement of the ball, respectively. In Sect. 4 we estimate the expected exit time: Theorem 4.1 provides (1.5) and Theorem 4.6 states (with more detail) the following estimates of the expected exit time of unimodal Lévy processes with unbounded Lévy–Khintchine exponent from $C^{1,1}$ open bounded sets $D$, under mild conditions including $(\mathbf{H})$,

$$\begin{aligned} \mathbb {E}^x\tau _{D}\approx V(\delta _{D}(x)), \quad x\in {\mathbb {R}^{d}}. \end{aligned}$$

In Sect. 5 we consider the case of transient $X$, and estimate the probability of ever hitting the ball from outside in, say, dimension $d\geqslant 3$, by using the estimates of Grzywny [24] for potential kernel: $U(x)\leqslant cV^2(|x|)/|x|^d$ for $x\in \mathbb {R}^d$, and for the capacity of the ball: ${{\mathrm{{ {Cap}}}}}(\overline{B_r})\approx r^d/V^2(r)$ for $r>0$. In Sect. 6 under weak scaling conditions we estimate the survival probability:

$$\begin{aligned} \mathbb {P}^x(\tau _{D}>t)\approx \frac{V(\delta _D(x))}{\sqrt{t}}\wedge 1,\qquad x\in {\mathbb {R}^{d}}, \quad 0<t\leqslant C V(r_0)^2. \end{aligned}$$

Here $r_0$ is the $C^{1,1}$-localization radius of $D$. The result is new even for complete subordinate Brownian motions. Further estimates and information are given as we proceed.

In Sect. 7 we discuss the role and validity of $(\mathbf{H})$ and give specific examples of Lévy processes manageable by our methods. Since $V(\delta _D(x))\approx [\psi (1/\delta _D(x))]^{-1/2}$, our estimates are often entirely explicit.

As we advance, the reader should observe the assumptions specified at the beginning of each section: as a rule they bind the statements of the results in that section. Notably, a large part of our estimates, especially of the upper bounds, are valid under minimal assumptions including isotropy and, usually but not always (cf. Sect. 2), unimodality of $X$. Scaling, unimodality, pure-jump character of $X$ and the Harnack-type condition $(\mathbf{H})$ on $V'$ are commonly assumed to prove matching lower bounds. We strive to make explicit the dependence of constants in our estimates on characteristics of $D$ and $X$. Some of the constants depend only on $d$ for all isotropic Lévy processes, others depend on the assumption of unimodality, the parameters in the weak scaling and other analytic properties of $X$ expressed through the Lévy measure. Good control of constants in estimates at scale $r>0$ necessitates the use of rather intrinsic quantities ${\mathcal {I}}(r)$ and ${\mathcal {J}}(r)$ introduced in Sect. 4. Such control is especially important for the study of unbounded sets.

2 Preliminaries

We write $f(x)\approx g(x)$ and say that $f$ and $g$ are comparable if $f, g\geqslant 0$ and there is a positive number $C$, called comparability constant, such that $C^{-1}f(x)\leqslant g(x)\leqslant C f(x)$ for all considered $x$. We write $C=C(a,\ldots ,z)$ to indicate that (constant) $C$ may be so chosen to depend only on $a,\ldots ,z$. Constant may change values from place to place except for capitalized numbered constants ($C_1$, $C_2$ etc.), which are the same at each occurrence.

We consider the Euclidean space $\mathbb {R}^d$ of arbitrary dimension $d\in \mathbb {N}$. All sets, functions and measures considered below are assumed Borel. Let $B(x,r)=\{y\in {\mathbb {R}^{d}}: |x-y|<r\}$, the open ball with center at $x\in {\mathbb {R}^{d}}$ and radius $r>0$, and let $B_r=B(0,r)$. We denote by $\omega _d=2\pi ^{d/2}/\Gamma (d/2)$ the surface measure of the unit sphere in ${\mathbb {R}^{d}}$. We also consider exterior sets $B^c(x,r)=(B(x,r))^c=\{y\in {\mathbb {R}^{d}}: |x-y|\geqslant r\}$, $B^c_r=(B(0,r))^c$ and $\overline{B}^c_r=(\overline{B(0,r)})^c$. For $D\subset {\mathbb {R}^{d}}$ we consider the distance to the complement of $D$:

$$\begin{aligned} \delta _D(x) ={{\mathrm{dist}}}(x,D^c),\qquad x\in {\mathbb {R}^{d}}. \end{aligned}$$

We say that $D$ is of class $C^{1,1}$ at scale $r$ if $r>0$, $D$ is open nonempty set in ${\mathbb {R}^{d}}$ and for every $Q\in \partial D$ there are balls $B(x',r)\subset D$ and $B(x'',r)\subset D^c$ tangent at $Q$. Thus, $B(x^{\prime }{},r)$ and $B(x^{\prime }{}^{\prime }{},r)$ are the inner and outer balls at $Q$, respectively. Estimates for $C^{1,1}$ open sets often rely on the inclusion $B(x^{\prime }{},r)\subset D\subset B(x^{\prime }{}^{\prime }{},r)^c$, domain monotonicity of the considered quantities and on explicit calculations for the extreme sides of the inclusion. If $D$ is $C^{1,1}$ at some unspecified scale (hence also at all smaller scales), then we simply say $D$ is $C^{1,1}$. The $C^{1,1}$-localization radius,

$$\begin{aligned} r_0=r_0(D)=\sup \{r: D \hbox { is } C^{1,1} \hbox { at scale } r\}, \end{aligned}$$

describes the local geometry of such $D$, while the diameter,

$$\begin{aligned} \mathrm{diam}(D)=\sup \{|x-y|:\;x,y\in D\}, \end{aligned}$$

depends on the global geometry of $D$. The ratio $\mathrm{diam}(D)/r_0(D)\geqslant 2$ is called the distortion of $D$. We remark that $C^{1,1}$ open sets may be defined by using local coordinates and Lipschitz condition on the gradient of the function defining their boundary (see, e.g., [1, Section 2]), hence the notation $C^{1,1}$. They can also be localized near the boundary without much changing the distortion [11, Lemma 1]. Some of the comparability constants in our estimates depend on $D$ only through $d$ and the distortion of $D$.

We denote by $C_c(D)$ the class of the continuous functions on ${\mathbb {R}^{d}}$ with compact support in (arbitrary) open $D\subset {\mathbb {R}^{d}}$, and we let $C_0(D)$ denote the closure of $C_c(D)$ in the supremum norm.

A Lévy process is a stochastic process $X=(X_t,\,t\geqslant 0)$ with values in ${\mathbb {R}^{d}}$, stochastically independent increments, cádlág paths and such that $\mathbb {P}(X(0)=0)=1$ [42]. We use $\mathbb {P}$ and $\mathbb {E}$ to denote the distribution and the expectation of $X$ on the space of cádlág paths $\omega :[0,\infty )\rightarrow {\mathbb {R}^{d}}$, in fact $X$ may be considered as the canonical map: $X_t(\omega )=\omega (t)$ for $t\geqslant 0$. In what follows, we shall use the Markovian setting for $X$, that is we define the distribution $\mathbb {P}^x$ and the expectation $\mathbb {E}^x$ for the Lévy process starting from arbitrary point $x\in {\mathbb {R}^{d}}$: $\mathbb {E}^xF(X)=\mathbb {E}F(x+X)$ for Borel functions $F\geqslant 0$ on paths. For $t\geqslant 0$, $x\in {\mathbb {R}^{d}}$, $f\in C_0({\mathbb {R}^{d}})$ we let $P_tf(x)=\mathbb {E}^x f(X_t)$, the semigroup of $X$. The distribution of $X_t$ under $\mathbb {E}=\mathbb {E}^0$ is denoted $p_t(dx)$ ($t\geqslant 0$) and forms a convolution semigroup of probability measures on ${\mathbb {R}^{d}}$.

We define the time of the first exit of $X$ from (Borel) $D\subset {\mathbb {R}^{d}}$:

$$\begin{aligned} \tau _D=\inf \{t>0: \, X_t\notin D\}. \end{aligned}$$

This random variable gives rise to a number of important objects in the potential theory of $X$. We shall focus on the expected exit time,

$$\begin{aligned} s_D(x)=\mathbb {E}^x \tau _D,\qquad x\in {\mathbb {R}^{d}}, \end{aligned}$$

(2.1)

and the survival probability

$$\begin{aligned} \mathbb {P}^x(\tau _D>t),\qquad x\in {\mathbb {R}^{d}},\,t>0. \end{aligned}$$

We shall also use the harmonic measure of $D$ for $X$ defined as

$$\begin{aligned} {\Omega }_D^x(A)=\mathbb {P}^x(X_{\tau _D}\in A), \qquad x\in {\mathbb {R}^{d}}, \quad A\subset {\mathbb {R}^{d}}. \end{aligned}$$

A real-valued function $f$ on ${\mathbb {R}^{d}}$ is called harmonic (for $X$) on open $D\subset {\mathbb {R}^{d}}$ if for every open $U$ such that $\overline{U}$ is a compact subset of $D$, we have

$$\begin{aligned} f(x)=\mathbb {E}^x f(X_{\tau _U})=\int _{U^c}f(y){\Omega }^x_U(dy),\quad x\in U, \end{aligned}$$

(2.2)

and the integral is absolutely convergent. In particular, if $g$ is defined on $D^c$, and $f(x)=\mathbb {E}^x g(X_{\tau _D})$ is absolutely convergent for $x\in D$, then $f$ is harmonic on $D$. This follows from the strong Markov property of $X$ [6]. A function $f$ is called regular harmonic in $D$ if (2.2) holds for $U=D$.

2.1 Isotropic Lévy processes

Lévy measure is a (nonnegative Borel) measure concentrated on ${\mathbb {R}^{d}{\setminus }\{0\}}$ such that

$$\begin{aligned} \int _{\mathbb {R}^{d}}\left( |x|^2\wedge 1\right) \nu (dx)<\infty . \end{aligned}$$

(2.3)

We call measure on ${\mathbb {R}^{d}}$ isotropic if it is invariant upon linear isometries of ${\mathbb {R}^{d}}$ (i.e. symmetric if $d=1$). A Lévy process $X_t$ [42] is called isotropic if all the measures $p_t(dx)$ are isotropic. Isotropic Lévy processes are characterized by Lévy–Khintchine (characteristic) exponents of the form

$$\begin{aligned} \psi (\xi )=\sigma ^2|\xi |^2+\int _{\mathbb {R}^{d}}\left( 1- \cos \langle \xi ,x\rangle \right) \nu (dx), \end{aligned}$$

(2.4)

with isotropic Lévy measure $\nu$ and $\sigma \geqslant 0$. To be specific, by the Lévy–Khintchine formula,

$$\begin{aligned} \mathbb {E}\,e^{i\left<\xi , X_t\right>}=\int _{\mathbb {R}^{d}}e^{i\left<\xi ,x\right>}p_t(dx)=e^{-t\psi (\xi )},\quad \xi \in {\mathbb {R}^{d}}. \end{aligned}$$

Unless explicitly stated otherwise, in what follows we assume that $X_t$ is an isotropic Lévy process in ${\mathbb {R}^{d}}$ with Lévy measure $\nu$ and characteristic exponent $\psi \not \equiv 0$ (we shall make additional assumptions in Sects. 2.2, 2.3 and 3). Since $\psi$ is a radial function, we shall often write $\psi (u)=\psi (x)$, where $x\in {\mathbb {R}^{d}}$ and $u=|x|\geqslant 0$. For the first coordinate $X_t^1$ of $X_t$ we obtain the same function $\psi (u)$. Clearly, $\psi (0)=0$ and $\psi (u)>0$ for $u>0$.

For $r>0$ we define, after [41],

$$\begin{aligned} \begin{array}{rcl} K(r)&{}=&{}\displaystyle \int \limits _{B_r}\frac{|z|^2}{r^2}\nu (dz), \quad L(r)= \nu \left( B^c_r\right) ,\\ h(r)&{}=&{}\displaystyle \frac{\sigma ^2 d}{r^2}+K(r)+L(r)=\frac{\sigma ^2 d}{r^2}+\int \limits _{{\mathbb {R}^{d}}}\left( \frac{|z|^2}{r^2}\wedge 1\right) \nu (dz). \end{array} \end{aligned}$$

(2.5)

We note that $0 \leqslant K(r)\leqslant h(r)<\infty$, $L(r)\geqslant 0$, $h$ is (strictly) positive and decreasing, and $L$ is non-increasing. The corresponding quantities for $X_t^1$, say $K_1(r)$, $L_1(r)$, $h_1(r)$, are given by the Lévy measure $\nu _1=\nu \circ x_1^{-1}$ on $\mathbb {R}$ [42, Proposition 11.10], in particular

$$\begin{aligned} h_1(r)=\frac{\sigma ^2}{r^2}+\int _\mathbb {R}\left( \frac{u^2}{r^2}\wedge 1\right) \nu _1(du)= \frac{\sigma ^2}{r^2}+ \int \limits _{{\mathbb {R}^{d}}}\left( \frac{|z_1|^2}{r^2}\wedge 1\right) \nu (dz),\quad r>0. \end{aligned}$$

We see that

$$\begin{aligned} h_1(r)\leqslant h(r)\leqslant h_1(r)d,\qquad r>0. \end{aligned}$$

(2.6)

We shall make connections to the expected exit time of $X$ for general open sets $D\subset {\mathbb {R}^{d}}$. By domain-monotonicity of exit times and Pruitt’s estimate (1.4), we have

$$\begin{aligned} s_D(x)\leqslant s_{B(x,\mathrm{diam}(D))}(x)\leqslant \frac{C}{h(\mathrm{diam}(D))}<\infty . \end{aligned}$$

(2.7)

Our first lemma is a slight improvement of [24, Lemma 3].

Lemma 2.1

If $r>0$ and $x\in B_{r/2}$, then $\mathbb {P}^x(|X_{\tau _{D}}|\geqslant r) \leqslant 24\, h(r)\, \mathbb {E}^x\tau _{D}$.

Proof

We assume that $\mathbb {E}^x \tau _D<\infty$, otherwise the result is trivial. Let $r>0$. Let ${\mathcal {A}}$ be the generator of the semigroup of $X$ acting on $C_0({\mathbb {R}^{d}})$. If $\phi \in C_{c}^2({\mathbb {R}^{d}})$, then $\phi$ is in the domain of ${\mathcal {A}}$, by Dynkin’s formula,

$$\begin{aligned} \mathbb {E}^x \int _0^{\tau _D} {\mathcal {A}} \phi (X_s)ds= \mathbb {E}^x\phi (X_{\tau _{D}})-\phi (x),\quad x\in {\mathbb {R}^{d}}, \end{aligned}$$

(2.8)

and the generator may be calculated pointwise as

$$\begin{aligned} {\mathcal {A}}\phi (x)&= {\sigma ^2\Delta \phi (x)+\int \left[ \phi (x+z)-\phi (x) -\mathbf{1}_{|z|<1}\left<z,\nabla \phi (x)\right>\right] \nu (dz)}. \end{aligned}$$

Since $\nu$ is symmetric, we can replace $\mathbf{1}_{|z|<1}$ in the above equation by $\mathbf{1}_{|z|<r}$. We shall use a function $g: {[0,\infty )}\mapsto {[-1,0]}$ such that $g(t)={-1}$ for $0\leqslant t\leqslant 1/2$, $g(t)= {0}$ for $t\geqslant 1$, and ${{\mathrm{ess\,sup}}}_{t{\geqslant 0} } |g'(t)|$ and ${{\mathrm{ess\,sup}}}_{t{\geqslant 0} } |g''(t)|$ are finite. In fact, we initially let $g''=16$ on $(1/2,3/4)$ and $g''=-16$ on $(3/4,1)$, which gives $\Vert g''\Vert _\infty =16$ and $\Vert g'\Vert _\infty =4$. We then have

$$\begin{aligned} 4\sup _{t{\geqslant 0} } |g'(t)| +\frac{1}{2}\sup _{t{\geqslant 0} }|g''(t)|&= 24,\end{aligned}$$

(2.9)

$$\begin{aligned} 2(d-1)\sup _t|g'(t)|+ \sup _t|g''(t)|&= 8(d+1). \end{aligned}$$

(2.10)

These will only slightly increase as we modify $g''$ to be continuous (such modified $g\in C^2$ is used below). Denote

$$\begin{aligned} {\phi _r}(y)=g(|y|/r), \quad y\in {\mathbb {R}^{d}}. \end{aligned}$$

We first consider $\phi _1$. Let $v,z\in {\mathbb {R}^{d}}$ be such that $|v+z|+|v|\geqslant 1/2$. There is a number $\theta$ between $|v|$ and $|v+z|$, such that

$$\begin{aligned} {\phi _1}(v+z)- {\phi _1}(v)&= g'(|v|)(|v+z|-|v|)+(1/2) g''(\theta )(|v+z|-|v|)^2\\&= g'(|v|)\frac{(|v+z|^2-|v|^2)}{|v+z|+|v|}+(1/2) g''(\theta )(|v+z|-|v|)^2\\&= g'(|v|)\frac{|z|^2+ 2\left<v,z\right>}{|v+z|+|v|}+(1/2) g''(\theta )(|v+z|-|v|)^2\\&= g'(|v|)\frac{ \left<v,z\right>}{|v|}+g'(|v|)\frac{ \left<v,z\right>}{|v|}\frac{|v|-|v+z|}{|v+z|+|v|}\\&+ g'(|v|)\frac{|z|^2}{|v+z|+|v|} +(1/2) g''(\theta )(|v+z|-|v|)^2. \end{aligned}$$

Since $|v+z|+|v|\geqslant 1/2$, we have

$$\begin{aligned} \left| g'(|v|)\frac{ \left<v,z\right>}{|v|}\frac{|v|-|v+z|}{|v+z|+|v|}\right| \leqslant |g'(|v|)|\frac{|z|^2}{|v+z|+|v|}\leqslant 2|g'(|v|)||z|^2. \end{aligned}$$

Also,

$$\begin{aligned} \frac{1}{2}g''(\theta )(|v+z|-|v|)^2\leqslant \frac{1}{2}|g''(\theta )||z|^2. \end{aligned}$$

Since

$$\begin{aligned} \left<z,\nabla {\phi _1}(v)\right>= g'(|v|)\frac{ \left<v,z\right>}{|v|},\end{aligned}$$

we obtain

$$\begin{aligned}\left| {\phi _1}(v+z)-{\phi _1}(v) -\mathbf{1}_{|z|<1}\left<z,\nabla {\phi _1}({v})\right>\right| \leqslant \left( 4\sup _t|g'(t)|+ \frac{1}{2}\sup _t|g''(t)|\right) |z|^2. \end{aligned}$$

If $|v+z|+|v|< 1/2$, then the latter inequality is also true because the left-hand side equals $0$. By changing variables we have

$$\begin{aligned} |{\phi _r}(v+z)-{\phi _r}(v) -\mathbf{1}_{|z|<r}\left<z,\nabla {\phi _r}(v)\right>| \leqslant \left( 4\sup _t|g'(t)|+ \frac{1}{2}\sup _t|g''(t)|\right) |z/r|^2. \end{aligned}$$

We also note that

$$\begin{aligned} |\Delta {\phi _1}(z)|=|(d-1)g'(|z|)/|z|+g''(|z|)|\leqslant 2(d-1)\sup _t|g'(t)|+ \sup _t|g''(t)|. \end{aligned}$$

Applying (2.8) to ${\phi _r}(y)=g(|y|/r)$ we get

$$\begin{aligned} \mathbb {P}^x\left( |X_{\tau _{D}}|\geqslant r\right) \leqslant \mathbb {E}^x{[\phi _r}(X_{\tau _{D}}){+1]}=\mathbb {E}^x \int _0^{\tau _D} {\mathcal {A}} {\phi _r}(X_s)ds, \qquad |x|\leqslant r/2. \end{aligned}$$

(2.11)

By the preceding estimates,

$$\begin{aligned} {\mathcal {A}} {\phi _r}(v)&= \sigma ^2\Delta {\phi _r}(v)+\int \left( {\phi _r}(v+z)-{\phi _r}(v) \nonumber -\mathbf{1}_{|z|<r}\left<z,\nabla {\phi _r}(x)\right>\right) \nu (dz)\\&\leqslant \sigma ^2\frac{\;2(d-1)\sup _t|g'(t)|+ \sup _t|g''(t)|}{r^2}\nonumber \\&+\frac{4\sup _t|g'(t)|+ \frac{1}{2}\sup _t|g''(t)| }{r^2}\int _{|z|<r}|z|^2\nu (dz) +\nu (B^c_{r}) . \end{aligned}$$

(2.12)

By (2.11), (2.12), (2.9), (2.10) and (2.5), we get the result. $\square$

Remark 1

The approach generalizes to other stopping times, e.g. deterministic times $t>0$:

$$\begin{aligned} \mathbb {P}^x\left( |X_t|\geqslant r\right) \leqslant 24\, h(r)\,t, \quad r>0,\quad |x|\leqslant r/2. \end{aligned}$$

(2.13)

Recall that $p_t(dx)$ has no atoms if and only if $\psi$ is unbounded (if and only if $\nu ({\mathbb {R}^{d}})=\infty$ or $\sigma >0$) [42, Theorem 30.10]. In fact, if $\sigma >0$ or if $d\geqslant 2$ and $\nu ({\mathbb {R}^{d}})=\infty$, then $(p_t,t>0)$ have lower semicontinuous density functions [49, (4.6)]. We further note that the resolvent measures

$$\begin{aligned} A\mapsto \int _0^\infty p_t(A)e^{-q t} dt, \qquad q>0, \end{aligned}$$

are absolutely continuous if and only if $p_t$, $t>0$, are absolutely continuous. This consequence of symmetry of $p_t$ is proved in [23, Theorem 6], see also [42, Remark 41.13].

2.2 Isotropic Lévy processes with unbounded characteristic exponent

Unless explicitly stated otherwise, in what follows $X$ is an isotropic Lévy process with unbounded Lévy–Khintchine exponent $\psi$.

Let $M_t=\sup _{s\leqslant t}X^1_s$ and let $L_t$ be the local time at $0$ for $M_t-X^1_t$, the first coordinate of $X$ reflected at the supremum [6, 22]. We consider its right-continuous inverse, $L^{-1}_s$, called the ascending ladder time process for $X^1_t$. We also define the ascending ladder-height process, $H_s = X_{L^{-1}_s}^{1} = M_{L^{-1}_s}$. The pair $(L^{-1}_t,H_t)$ is a two-dimensional subordinator [6, 22]. In fact, since $X_t^{1}$ is symmetric and has infinite Lévy measure or nonzero Gaussian part, by [22, Corollary 9.7], the Laplace exponent of $(L^{-1}_t,H_t)$ is

$$\begin{aligned} {-\frac{1}{t}}\log \left( \mathbb {E}\exp [-\tau L_t^{-1}-\xi H_t]\right) =c_+\exp \left\{ \frac{1}{\pi }\int _0^\infty \frac{\log \left[ \tau +\psi (\theta \xi )\right] }{1+\theta ^2} d\theta \right\} ,\quad \tau ,\xi \geqslant 0, \end{aligned}$$

In what follows we let $c_+=1$, thus normalizing the local time $L$ [22]. In particular, $L^{-1}_s$ is then the standard $1/2$-stable subordinator (see also [21, (4.4.1)]), and the Laplace exponent of $H_t$ is

$$\begin{aligned} \kappa (\xi )= {-\frac{1}{t}}\log \left( \mathbb {E}\exp [-\xi H_t]\right) = \exp \left\{ \frac{1}{\pi } \int _0^\infty \frac{ \log {\psi }(\xi \zeta )}{1 + \zeta ^2} \, d\zeta \right\} , \quad \xi \geqslant 0. \end{aligned}$$

(2.14)

The renewal function $V$ of the ascending ladder-height process $H$ is defined as

$$\begin{aligned} V(x) = \int _0^{\infty }\mathbb {P}(H_s \leqslant x)ds, \quad x {\in {\mathbb {R}}}. \end{aligned}$$

(2.15)

Thus, $V(x)=0$ if $x<0$ and $V$ is non-decreasing. It is also well known that $V$ is subadditive,

$$\begin{aligned} V(x+y)\leqslant V(x)+V(y), \quad x,y \in \mathbb {R}, \end{aligned}$$

(2.16)

and $V(\infty )=\infty$. Furthermore, from (2.14) we infer that the Laplace transform ${\mathcal {L}}V$ of $V$ is

$$\begin{aligned} {\mathcal {L}}V(\xi )= \frac{1}{ \xi \kappa (\xi )}. \end{aligned}$$

(2.17)

Both $V$ and its derivative $V'$ play a crucial role in our development. They were studied by Silverstein as $g$ and $\psi$ in [45], see (1.8) and Theorem 2 ibid., respectively. If resolvent measures of $X^1_t$ are absolutely continuous, then it follows from [45, Theorem 2] that $V(x)$ is absolutely continuous and harmonic on $(0,\infty )$ for the process $X_t^1$, in fact, $V$ is invariant for the process $X_t^1$ killed on exiting $(0,\infty )$. Also, $V^\prime$ is a positive harmonic function for $X_t^1$ on $(0,\infty )$, hence $V$ is actually (strictly) increasing. Notably, the definition of $V$ is rather implicit and the study of $V$ poses problems. In fact, we shall shortly present sharp estimates of $V$ by means of (simpler) functions $\psi$ and $h$, but decay properties of $V'$ are more delicate and they are not yet fully understood. Under structure assumptions satisfied for complete subordinate Brownian motions, $V'$ is monotone, in fact completely monotone (cf. Lemma 7.5). This circumstance stimulated much of the progress made in [17, 36]. The methods presented below in this paper address more general situations, e.g. when the Lévy–Khintchine exponent $\psi$ has weak scaling or when $X$ has a nonzero Gaussian part (see Sect. 7.1).

By [21, Corollary 4 and Theorem 3] and [44, Remark 3.3 (iv)] the following result holds.

Lemma 2.2

We have $\lim _{\xi \rightarrow \infty }\kappa (\xi )/\xi =\sigma$. Furthermore, if $\sigma >0$, then $V'$ is continuous, positive and bounded by $\lim _{t\rightarrow 0^+}V'(t)=\sigma ^{-1}$. In fact $V'$ is bounded if and only if $\sigma >0$.

As we indicated in Sect. 1, estimates of $\mathbb {E}^x\tau _{B_{r}}$, the expected exit time from the ball play an important role in this paper. The upper bound (1.4), sharp at the center of the ball, was given by Pruitt in [41, p. 954]. It was later generalized to more general Markov processes by Schilling in [43, Remark 4.8]. For every symmetric Lévy process $X$ on $\mathbb {R}^1$ with unbounded Lévy–Khintchine exponent $\psi$, the following bound with absolute constant $C_0>0$ follows from [26, Proposition 3.5] by Grzywny and Ryznar and subadditivity of $V$,

$$\begin{aligned} {C_0}V(r)V(r-|x|) \leqslant \mathbb {E}^x\tau _{(-r,r)}\leqslant {2}V(r)V(r-|x|), \qquad x\in \mathbb {R},\quad r>0. \end{aligned}$$

(2.18)

In Sect. 4 we establish a similar comparability result in arbitrary dimension under appropriate conditions on $X$. The upper bound is, however, simpler, and we can give it immediately.

Lemma 2.3

For all $r>0$ and $x\in {\mathbb {R}^{d}}$ we have ${\mathbb {E}}^x\tau _{B_r}\leqslant 2 V(r)V(r-|x|)$.

Proof

Since $X$ is isotropic with unbounded Lévy–Khintchine exponent $\psi$, by Blumenthal’s 0-1 law we have $\tau _{B_r}=0$ $\mathbb {P}^x$-a.s. for all $x\in B^c_r$. Hence, it remains to prove the claim for $x\in B_r$. If $\tau =\inf \{t>0: |X^1_t|>r\}$, then domain-monotonicity of the exit times and [26, Proposition 3.5] yield $\mathbb {E}^x\tau _{B_r}\leqslant \mathbb {E}^x\tau \leqslant V(r-|x_{1} |)V(2r)$. By (2.16) and rotations we obtain the claim. $\square$

We define the maximal characteristic function $\psi ^*(u):= {\sup _{0\leqslant s\leqslant {u}}\psi (s)}$, where $u\geqslant 0$.

Proposition 2.4

The constants in the following comparisons depend only on the dimension,

$$\begin{aligned} h(r)\approx h_1(r)\approx \psi ^*(1/r)\approx \left[ V(r)\right] ^{-2} ,\quad r>0. \end{aligned}$$

(2.19)

Proof

We shall see that all of the comparisons are absolute, except for the first comparison in (2.19), which depends on the dimension via (2.6). Let $r>0$. Since $X^1$ is symmetric,

$$\begin{aligned} h_1(r)\approx \psi ^*(1/r), \end{aligned}$$

(2.20)

see [24, Corollary 1]. Let $r>0$ and $\tau _r$ be the time of the first exit of $X^1_t$ from the interval $(-r,r)$. By (2.18) and (1.4), we have $V^2(r)\approx \mathbb {E}^0\tau _r \approx 1/h_1(r)$. $\square$

Lemma 2.5

We have $\lim _{t\rightarrow 0^+}t/V(t)=\sigma .$

Proof

By Proposition 2.4 and the dominated convergence theorem,

$$\begin{aligned} \frac{t^2}{V^2(t)}\approx t^2h_1(t)= \sigma ^2+\int _{\mathbb {R}^d} \left( t^2\wedge |z_1|^2\right) \nu (dz)\rightarrow \sigma ^2 \quad \text { as }t\rightarrow 0. \end{aligned}$$

This ends the proof when $X$ is pure-jump. If $\sigma >0$, then we use Lemma 2.2. $\square$

The next result on survival probability was known before in the situation when $\psi (r)$ and ${r^2}/{\psi (r)}$ are non-decreasing in $r\in (0, \infty )$, see [37, Theorem 4.6].

Proposition 2.6

For every symmetric Lévy process in $\mathbb {R}$ which is not compound Poisson,

$$\begin{aligned} \mathbb {P}^x(\tau _{(0, \infty )}\geqslant t)\approx 1\wedge \frac{1}{\sqrt{t\psi ^*(1/x)}},\quad t,x>0, \end{aligned}$$

(2.21)

and the comparability constant is absolute.

Proof

Considering that $L^{-1}_s$ is a $1/2$-stable subordinator, from [37, Theorem 3.1] we see that

$$\begin{aligned} \mathbb {P}^x(\tau _{(0, \infty )}\geqslant t)\approx 1\wedge \frac{V(x)}{\sqrt{t}}, \qquad {t,\, x >0.} \end{aligned}$$

(2.22)

The result now obtains from (2.22) and Proposition 2.4. $\square$

Remark 2

If $\Pi \subset {\mathbb {R}^{d}}$ is an open halfspace, $R>0$ and $B_R\subset \Pi$, then $\mathbb {P}^x(\tau _{B_R}>t)\leqslant \mathbb {P}^x(\tau _{\Pi }>t)$ and $\mathbb {P}^x(\tau _{\overline{B}^c_R}>t)\geqslant \mathbb {P}^x(\tau _{\overline{\Pi }^c}>t)$. By (2.22), for all $t>0$ and $x\in {\mathbb {R}^{d}}$ we obtain

$$\begin{aligned} \mathbb {P}^x(\tau _{B_R}>t)\leqslant C \left( 1\wedge \frac{V(\delta _{B_R}(x))}{\sqrt{t}}\right) \end{aligned}$$

and

$$\begin{aligned} \mathbb {P}^x(\tau _{\overline{B}^c_R}>t)\geqslant C^{-1} \left( 1\wedge \frac{V(\delta _{\overline{B}_R^c}(x))}{\sqrt{t}}\right) , \end{aligned}$$

where $C$ is an absolute constant. Namely, we let $\partial \Pi$ touch $\partial B_R$ at the point closest to $x$. The inequalities hold for all isotropic Lévy processes which are not compound Poisson.

From (2.19) and definitions of $L_1$, $L$ and $h$, we derive the following inequality,

$$\begin{aligned} L_1(r)\leqslant L(r)\leqslant h(r)\leqslant c/[V(r)]^2, \qquad r>0. \end{aligned}$$

(2.23)

Lemma 2.7

There is $C_1=C_1(d)$ such that if $r>0$, $D\subset B_r$ and $x\in {D\cap } B_{r/2}$, then

$$\begin{aligned} \mathbb {P}^x\left( |X_t|\geqslant r\right)&\leqslant C_1 \frac{ t}{V^2(r)}, \quad t>0,\end{aligned}$$

(2.24)

$$\begin{aligned} \mathbb {P}^x\left( |X_{\tau _{D}}|\geqslant r\right)&\leqslant C_1 \frac{ \mathbb {E}^x\tau _{D}}{V^2(r)}, \quad \text {and}\end{aligned}$$

(2.25)

$$\begin{aligned} \mathbb {E}^x\tau _{B_{r}}&\geqslant V^2(r)/C_1. \end{aligned}$$

(2.26)

Proof

Lemma 2.1, Proposition 2.4 and (2.13) give (2.24) and (2.25), which yield (2.26). $\square$

Corollary 2.8

There exist $C_2=C_2(d)$ and $C_3=C_3(d)$ such that for $t,r> 0$ and $|x|\leqslant r/2$,

$$\begin{aligned} \mathbb {P}^x(\tau _{B_r}\leqslant t )\leqslant C_2\frac{t}{V^2(r)}, \end{aligned}$$

and

$$\begin{aligned} \mathbb {P}^x(\tau _{B_r}>C_3V^2(r) )\geqslant 1/2. \end{aligned}$$

Proof

Observe that for $|x|\leqslant r/2$,

$$\begin{aligned} \mathbb {P}^x(\tau _{B_r}\leqslant t )\leqslant {\mathbb {P}}^0(\tau _{B_{r/2}}\leqslant t ). \end{aligned}$$

By Lévy’s inequality and (2.24) we obtain the first claim with $C_2=8C_1$, because

$$\begin{aligned} \mathbb {P}^0(\tau _{B_{r/2}}\leqslant t )&= \mathbb {P}^0\left( \sup _{s\leqslant t}|X_s|\geqslant r/2\right) \\ {}&\leqslant 2 \mathbb {P}^0( |X_t|\geqslant r/2)\leqslant 2C_1\frac{t}{V^2(r/2)}\\&\leqslant 8C_1\frac{t}{V^2(r)}. \end{aligned}$$

Taking $t= V^2(r)/(16C_1)$ we prove the second claim with $C_3=(16 C_1)^{-1}$. $\square$

We observe the following regularity of the expected exit time.

Lemma 2.9

If the resolvent measures of $X$ are absolutely continuous and the open bounded set $D\subset {\mathbb {R}^{d}}$ has the outer cone property, then $s_D\in C_0(D)$.

Proof

Recall that $s_D$ is bounded. We also have $s_D(x)=0$ for $x\in D^c$. Indeed, for $x\in \partial D$, by Blumenthal’s 0-1 law we have $\tau _D=0$ $\mathbb {P}^x$-a.s., because $X$ is isotropic with unbounded Lévy–Khintchine exponent $\psi$ and $D$ has the outer cone property. Due to [23, Theorem 6] and [27, Lemma 2.1], $X$ is strong Feller. Hence, for each $t>0$, $x\mapsto \mathbb {P}^x(\tau _D>t)$ is upper semicontinuous [19, Proposition 4.4.1, p. 163]. Therefore $s_D(x)=\int _0^\infty \mathbb {P}^x(\tau _D>t)dt$ is also upper semi-continuous. In consequence, $s_D(x)\rightarrow 0$ as $\delta _D(x)\rightarrow 0$, and so $s_D$ is continuous at $\partial D$. To prove continuity of $s_D$ on $D$, we let $D\ni z\rightarrow x\in D$, and denote

$$\begin{aligned} D'=D-(z-x), \quad U=D\cap D', \quad R=D{\setminus }U. \end{aligned}$$

We have $s_D(x)=s_U(x)+\int _R s_D(y)\omega ^x_U(dy)$ and $s_D(z)=s_{D'}(x)\geqslant s_U(x)$, thus $s_D(z)\geqslant s_D(x)-\int _{R} s_D(y)\omega ^x_U(dy)\rightarrow s_D(x)$, because if $y\in R$, then $\delta _D(y)\leqslant |z-x|$ and $s_D(y)$ is small. We see that $s_D$ is lower semi-continuous on $D$, hence continuous in $D$, in fact on ${\mathbb {R}^{d}}$. $\square$

Remark 3

The resolvent measures are absolutely continuous in dimensions bigger than one, hence $s_D\in C_0(D)$ if $D$ is an open bounded set with the outer cone property in $\mathbb {R}^d$ and $d\geqslant 2$. This is also the case in dimension $d=1$ under the assumptions of the next section.

2.3 Isotropic absolutely continuous Lévy measure

In what follows, unless stated otherwise, we assume that $X$ is an isotropic Lévy process in ${\mathbb {R}^{d}}$ with the Lévy measure $\nu (dx)=\nu (x)dx$ and unbounded Lévy–Khintchine exponent $\psi$. In particular, $X$ is symmetric, not compound Poisson, has absolute continuous distribution for all $t>0$ and absolutely continuous resolvent measures. Indeed, the case of $d\geqslant 2$ was discussed in Sect. 2.1 and Remark 3, and for $d=1$ we invoke [46, Theorem 1 (i)(ii)]. We may assume that the density functions $x\mapsto p_t(x)$ are lower-semicontinuous for every $t>0$, see [27, Theorem 2.2].

The transition density of the process $X$ killed off open $D{\subset {\mathbb {R}^{d}}}$ is defined by Hunt’s formula,

$$\begin{aligned} p_D(t,x,y) =p(t,x,y)-\mathbb {E}^x\big [p(t-\tau _D,X_{\tau _D},y);\tau _D<t\big ], \qquad t>0, \, x,y\in {\mathbb {R}^{d}}. \end{aligned}$$

We call $p_D$ the Dirichlet heat kernel of $X$ on $D$. The Green function of $D$ for $X$ is defined as

$$\begin{aligned} G_D(x,y)=\int _0^\infty p_D(t,x,y)dt. \end{aligned}$$

Here is a connection between the main objects of our study,

$$\begin{aligned} s_D(x)=\mathbb {E}^x \tau _D=\int _{\mathbb {R}^{d}}G_D(x,y)dy=\int _0^\infty \mathbb {P}^x(\tau _D>t)dt. \end{aligned}$$

(2.27)

If $x \in D$, then the $\mathbb {P}^x$-distribution of $(\tau _D,X_{\tau _D-},X_{\tau _D})$ restricted to $X_{\tau _D-}\ne X_{\tau _D}$ is given by the following density function [29],

$$\begin{aligned} (0,\infty )\times D\times \left( \overline{D}\right) ^c\ni (s, u, z) \mapsto \nu (z -u) p_D(s, x, u). \end{aligned}$$

(2.28)

Integrating against $ds$, $du$ and/or $dz$ gives marginal distributions. For instance, if $x\in D$ then

$$\begin{aligned} \mathbb {P}^x(X_{\tau _D}\in dz)=\left( \int _D G_D(x,u) \nu (z-u) du\right) dz \quad \text { on } \;(\overline{ D})^c. \end{aligned}$$

(2.29)

Identities resulting from (2.28) are called Ikeda–Watanabe formulae for $X$. Noteworthy, they allow for intuitive interpretations in terms of the expected occupation time measures $p_D(s,x,u)du$ and $G_D(x,u)du$, and in terms of the measure of the intensity of jumps, $\nu (z-u)dz$, cf. [7, p. 17].

3 Barriers for unimodal Lévy processes

A measure on ${\mathbb {R}^{d}}$ is called isotropic unimodal, in short, unimodal, if it is absolutely continuous on ${\mathbb {R}^{d}}{\setminus }\{0\}$ with a radial non-increasing density function (such measures may have an atom at the origin). A Lévy process $X_t$ is called (isotropic) unimodal if the distributions $p_t(dx)$ are unimodal. Unimodal Lévy processes are characterized in [48] by unimodal Lévy measures $\nu (dx)=\nu (x)dx=\nu (|x|)dx$. For the unimodal process, $p_t(dx)$ has a radial nonincreasing density $p_t(x)$ on ${\mathbb {R}^{d}{\setminus }\{0\}}$, and an atom at the origin, with mass $\exp [-t\nu ({\mathbb {R}^{d}})]$ (no atom if $\psi$ is unbounded, i.e. if $\sigma >0$ or $\nu ({\mathbb {R}^{d}})=\infty$). We refer to [10] for additional discussion. Unless explicitly stated otherwise, in what follows we always assume that $X$ is a unimodal Lévy process in ${\mathbb {R}^{d}}$ with unbounded Lévy–Khintchine exponent $\psi$. Recall that by [10, Proposition 2],

$$\begin{aligned} \psi (u)\leqslant \psi ^*(u)\leqslant \pi ^2\, \psi (u) \quad \text { for } \ u\geqslant 0. \end{aligned}$$

(3.1)

For $f:{\mathbb {R}^{d}}\rightarrow \mathbb {R}$, $t>0$ and $x\in {\mathbb {R}^{d}}$ we consider the (approximating) Dynkin operator,

$$\begin{aligned} {\mathcal {A}}_t f(x)= \frac{\mathbb {E}^xf(X_{\tau _{B(x,t)}})-f(x)}{\mathbb {E}^x\tau _{B(x,t)}}, \end{aligned}$$

whenever $\mathbb {E}^xf(X_{\tau _{B(x,t)}})$ is well defined. For instance, if $s_D(x)=\mathbb {E}^x \tau _D$ and $0<t\leqslant \delta _D(x)$, then by the strong Markov property, $s_D(x)=s_{B(x,t)}(x)+\mathbb {E}^x s_D(X_{\tau _{B(x,t)}})$, and so

$$\begin{aligned} {\mathcal {A}}_t s_D(x)=-1. \end{aligned}$$

(3.2)

By a similar argument, if $f$ is harmonic on $D$, $x\in D$ and $0<t<\delta _D(x)$, then ${\mathcal {A}}_t f(x)= 0$, by the (harmonic) mean-value property. In particular, let $\mathbb {H}=\{x=(x_1,\ldots ,x_d)\in {\mathbb {R}^{d}}:\; x_1>0\}$ and $V_1(x)=V(x_1)$. Since $V$ is harmonic on $(0,\infty )\subset \mathbb {R}$ for $X_1$, $V_1$ is harmonic in $\mathbb {H}$ for $X$ and so ${\mathcal {A}}_t V_{1}(x)=0$, if $0<t<\delta _\mathbb {H}(x)$ (this is the main reason why $V$ is relevant for the construction of barriers for $C^{1,1}$ sets in ${\mathbb {R}^{d}}$). We also observe the following minimum principle: if $x$ is a point in ${\mathbb {R}^{d}}$ and $f(x)=\inf _{y\in {\mathbb {R}^{d}}}{f(y)}$, then ${\mathcal {A}}_t f(x)\geqslant 0$ for every $t>0$.

Corollary 3.1

If ${\mathcal {A}}_t f(x)<0$ for some $t>0$, then $f(x)>\inf _{y\in {\mathbb {R}^{d}}}f(y)$.

Lemma 3.2

If $f\in C_0(D)$ and for every $x\in D$ there is $t>0$ such that ${\mathcal {A}}_t f(x)<0$, then $f\geqslant 0$ on ${\mathbb {R}^{d}}$.

Proof

Since $f$ attains its infimum on ${\mathbb {R}^{d}}$, but not on $D$ (cf. Corollary 3.1), we have $f\geqslant 0$. $\square$

We make a simple observation on local regularity of harmonic functions, motivated by [12, proof of Lemma 6] (see [5, 47] for more in this direction).

Lemma 3.3

Let $X$ be an isotropic Lévy process with absolutely continuous Lévy measure. If $g$ is bounded on ${\mathbb {R}^{d}}$ and harmonic on open $D\subset {\mathbb {R}^{d}}$, then $g$ is continuous on $D$.

Proof

For $r>0$, let ${\Omega }_r(dy)=\mathbb {P}^0(X(\tau _{B_r})\in dy)$. Note that $g(x)=\int _{\mathbb {R}^{d}}g(y+x){\Omega }_r(dy)$ if $0<r<\delta _D(x)$. By the isotropy and Ikeda–Watanabe formula, ${\Omega }_r(dy)=c_r\sigma _r(dy)\,+\,\phi _r(y)dy$, where $\sigma _r$ is the normalized spherical measure on $\partial B_r$, $0\!\leqslant \! c_r\!\leqslant \! 1$, and

$$\begin{aligned} {\phi _r}(y)= \left\{ \begin{array}{ll} \int _{B_r} G_{B_r}(0,v)\nu (y-v)dv, &{} \text { if } |y|>r,\\ 0&{} \text { else. } \end{array}\right. \end{aligned}$$

Let $\rho \,{>}\,0$. Note that the measure $\int _\rho ^{2\rho }c_r\sigma _r(A) dr$ has density function $\omega _d^{-1}\mathbf{1}_{\rho <|x|<2\rho }\;|x|^{1-d}\,c_{|x|}$. Therefore ${\Phi }_\rho (A)={\rho ^{-1}}\int _\rho ^{2\rho }{\Omega }_r(A) dr$ is absolutely continuous, with density function denoted $F_\rho$. We have $g(x)=\int _{\mathbb {R}^{d}}g(y+x){\Phi }_\rho (dy)=\int _{\mathbb {R}^{d}}g(y)F_\rho (y-x){dy}$ if $\delta _D(x)>2\rho$. So, locally on $D$, $g$ is a convolution of a bounded function with an integrable function, so it is continuous on $D$. $\square$

Lemma 3.4

Let $D\subset {\mathbb {R}^{d}}$ be an open bounded set with the outer cone property and let function $f$ be continuous on ${\mathbb {R}^{d}}$. If $\int _{B_r^c}|f(y)|\nu (y/2)dy<\infty$ for some $r>0$, then $g(x)=\mathbb {E}^xf(X_{\tau _D})$ is continuous on ${\mathbb {R}^{d}}$.

Proof

Since $D$ has the outer cone property, $f(x)=g(x)$ for all $x\in D^c$, and so $g$ is continuous as a function on $D^c$. To prove the continuity of $g$ on $\overline{D}$, we first notice that $f$ is locally bounded, hence $\int _{B_r^c}|f(y)|\nu (y/2)dy<\infty$ for all  $r>0$. Let $r_1>0$ be such that $D\subset B_{r_1/2}$, and let $r\geqslant r_1$. We have

$$\begin{aligned} g(x)&=\mathbb {E}^x f(X_{\tau _D})\mathbf{1}_{B_r}(X_{\tau _D})+ \mathbb {E}^x f(X_{\tau _D})\mathbf{1}_{B^c_r}(X_{\tau _D}), \quad x\in {\mathbb {R}^{d}}. \end{aligned}$$

(3.3)

For clarity, the second term in (3.3) is finite because by the Ikeda–Watanabe formula,

$$\begin{aligned} \int _{B_r^c}|f(y)|\int _{D}G_D(x,z)\nu (y-z)dzdy\leqslant \mathbb {E}^x\tau _D \int _{B_r^c}|f(y)|\nu (y/2)dy<\infty . \end{aligned}$$

(3.4)

It follows that $g$ is well defined and finite everywhere. Of course, $g$ and each term in (3.3) are (regular) harmonic in $D$. Let $x_0\in \partial D$, $0<\rho <r_1/2$ and $|x-x_0|<\rho /2$. We have

$$\begin{aligned} g(x)-g(x_0)&=\mathbb {E}^x\left[ f(X_{\tau _D})-f(x_0)\right] \mathbf{1}_{B(x_0,\rho )}(X_{\tau _D})\\&\quad +\mathbb {E}^x\left[ f(X_{\tau _D})-f(x_0)\right] \mathbf{1}_{B_r\setminus B(x_0,\rho )}(X_{\tau _D})\\&\quad +\mathbb {E}^x \left[ f(X_{\tau _D})-f(x_0)\right] \mathbf{1}_{B^c_r}(X_{\tau _D}). \end{aligned}$$

By Lemma 2.1 and (3.4),

$$\begin{aligned} |g(x)-g(x_0)|&\leqslant \sup _{y\in B(x_0,\rho )}|f(y)-f(x_0)| +48 h(\rho ) \mathbb {E}^x\tau _{D}\sup _{y\in B_r} |f(y)|\\&\quad + \mathbb {E}^x\tau _D \int _{B_r^c}|f(y)-f(x_0)|\nu (y/2)dy. \end{aligned}$$

By this and Remark 3 we see that $g(x)\rightarrow g(x_0)$ as $x\rightarrow x_0$.

By (3.3), Lemma 3.3 and (3.4) we see that locally on $D$, $g$ is a uniform limit of continuous functions. Thus, $g$ is continuous on ${\mathbb {R}^{d}}$. $\square$

Lemma 3.5

There is $c=c(d)$ such that for every $r>0$,

$$\begin{aligned} \int _{B_r^c}V(|y|)\nu (y)dy\leqslant \frac{c}{V(r)}. \end{aligned}$$

Proof

Recall that $L(r)= \nu \left( B^c_r\right)$. Integration by parts and (2.23) yield

$$\begin{aligned} \int _{B_{r}^c}V(|y|)\nu (y)dy&= \int _{r}^\infty \!\!\omega _d\,\nu (\rho )\rho ^{d-1}V(\rho )d\rho = L(r)V(r)+\int _{r}^\infty V'(\rho )L(\rho )d\rho \\&\leqslant c/V(r)+c\int _{r}^\infty V'(\rho )/V^2(\rho )d\rho \leqslant c/V(r), \end{aligned}$$

where $c=c(d)$. $\square$

Our main goal in the remainder of this section is to approximate harmonic functions of $X$ in the ball and the complement of the ball. We start with estimates of auxiliary integrals.

Recall that $h$ is defined in (2.5). In (3.5) below we make an important observation on $h'$.

Proposition 3.6

There is $C=C(d)$ such that

$$\begin{aligned} \int _{0}^r {V(\rho ) } \rho ^d\nu (\rho )d \rho&\leqslant C \frac{r}{V(r)} ,\quad r>0. \end{aligned}$$

Proof

Recall that $K(u)=\omega _du^{-2}\int ^{u}_0 \rho ^{d+1}\nu (\rho )d\rho$, $L(u)=\omega _d\int _{u}^\infty \rho ^{d-1}\nu (\rho )d\rho$, and $h(u)=K(u)+L(u)+u^{-2}\sigma ^2 d$. Since $\nu$ is non-increasing, hence a.e. continuous, for a.e. $u\in \mathbb {R}$ we have

$$\begin{aligned} h'(u)&= -2u^{-1}K(u)+\omega _du^{d-1}\nu (u) - \omega _du^{d-1}\nu (u)-2u^{-3}\sigma ^2 d\nonumber \\&= -2u^{-1}\left( K(u)+u^{-2}\sigma ^2 d\right) . \end{aligned}$$

(3.5)

Also,

$$\begin{aligned} \int _{0}^r {V(\rho ) } \rho ^d\nu (\rho )d \rho \leqslant c_1 \int _0^{r/2} V(u)L(u) d u, \end{aligned}$$

(3.6)

because

$$\begin{aligned} \rho ^d\nu (\rho )&= \frac{d2^d}{2^d-1}\int ^{\rho }_{\rho /2}u^{d-1}\nu (\rho )du \leqslant \frac{d2^d}{2^d{-}1}\int ^{\rho }_{\rho /2}u^{d{-}1}\nu (u)du\leqslant \frac{d2^d}{\omega _d(2^d-1)}L(\rho /2). \end{aligned}$$

By (2.19), $V(u)\approx h^{-1/2}(u)$, and so (3.5) yields

$$\begin{aligned} V(u)L(u)&\approx h^{-1/2}(u)(h(u)-K(u)-u^{-2}\sigma ^2 d)=h^{-1/2}(u)\left( h(u)+\frac{u}{2}h'(u)\right) \\&= (uh^{1/2}(u))' \quad a.e. \end{aligned}$$

From this and (3.6) we obtain the result

$$\begin{aligned} \int _{0}^r {V(\rho ) } \rho ^d\nu (\rho )d \rho&\leqslant c_2 rh^{1/2}(r)\approx \frac{r}{V(r)}. \end{aligned}$$

$\square$

Lemma 3.7

There exists a constant $C=C(d)$, such that for $0<x<r$,

$$\begin{aligned} \int _0^{r}V'(y/2)\int _{ |y-x|}^r \rho ^d\nu (\rho /2)d \rho dy\leqslant C\frac{r}{V(r)}, \end{aligned}$$

and

$$\begin{aligned} \int _0^{r} V'(y/2) {|y-x|^{d+1}} \nu (|y-x|/2) dy\leqslant C\frac{r}{V(r)}. \end{aligned}$$

Proof

Since $\nu$ is decreasing, we have

$$\begin{aligned} {|y-x|^{d+1}} \nu (|y-x|{/2})\leqslant {2(d+1)}\int _{ |y-x|{/2}}^{|y-x|} \rho ^d\nu (\rho /2)d\rho , \end{aligned}$$

hence

$$\begin{aligned} \int _0^{r} V'(y/2) {|y-x|^{d+1}} \nu (|y-x|{/2}) dy&\leqslant {2(d+1)}\int _0^{r}V'(y/2)\int _{ |y-x|{/2}}^{{r}} \rho ^d\nu (\rho /2)d \rho dy. \end{aligned}$$

To completely prove the lemma it is enough to estimate the latter integral. It equals

$$\begin{aligned}&2\int _0^r \rho ^d \nu (\rho /2)\int _{(x/2-\rho )\vee 0}^{(x/2+\rho )\wedge r/2}V'(z)dzd\rho \\&\quad \leqslant 2\int _0^r \rho ^d \nu (\rho /2)[V(x/2+\rho )-V(x/2-\rho )]d\rho \\&\quad \leqslant 4\int _0^r \rho ^d \nu (\rho /2)V(\rho ) d\rho \leqslant cr/V(r), \end{aligned}$$

where we used subadditivity (2.16) of $V$ on $\mathbb {R}$ and Proposition 3.6. $\square$

Recall that $V>0$ and $V'>0$ on $(0,\infty )$.

Definition 1

We say that condition $(\mathbf{H})$ holds if for every $r>0$ there is $H_r\geqslant 1$ such that

$$\begin{aligned} V(z)-V(y)\leqslant H_r \,V^\prime (x)(z-y)\quad \text {whenever}\quad 0<x\leqslant y\leqslant z\leqslant 5x\leqslant 5r. \end{aligned}$$

(3.7)

We say that $(\mathbf{H^*})$ holds if $H_\infty =\sup _{r>0}H_r<\infty$.

We consider $(\mathbf{H})$ and $(\mathbf{H^*})$ as Harnack type because $(\mathbf{H})$ is implied by the following property:

$$\begin{aligned} \sup _{x\leqslant r,y\in [x,5x]} \!V'(y)\leqslant \ H_r \inf _{x\leqslant r,y\in [x,5x]}\!V'(y), \qquad r>0. \end{aligned}$$

(3.8)

Both conditions control relative growth of $V$. If $(\mathbf{H})$ holds, then we may and do chose $H_r$ non-decreasing in $r$. Each of the following situations imply $(\mathbf{H})$:

• 1. $X$ is a subordinate Brownian motion governed by a special subordinator (see Lemma 7.5).

• 2. $d\geqslant 3$ and the characteristic exponent of $X$ satisfies WLSC (see (5.1) and Lemma 7.2).

• 3. $d\geqslant 1$ and the characteristic exponent of $X$ satisfies WLSC and WUSC (see (5.2) and Lemma 7.3).

• 4. $\sigma >0$ in (2.4) (see Lemma 7.4).

A more detailed discussion of $(\mathbf{H})$ and further examples are given in Sect. 7.

The following Lemma  3.8 and Lemma 3.9 are the main results of this section. They exhibit nonnegative functions which are superharmonic (hence barriers) or subharmonic near the boundary of the ball, inside or outside of the ball, respectively. The functions are obtained by composing $V$ with the distance to the complement of the ball or to the ball, respectively. Super- and subharmonicity are defined by the left-hand side inequality in (3.9) and (3.16), respectively. The super- and subharmonicity of the considered functions are relatively mild as we have good control of the right-hand sides of these inequalities (see the proof of Theorem 4.1 for an application). In comparison with previous developments, it is the use of Dynkin’s operator that allows for calculations which only minimally depend on the differential regularity of $V$ (the dependence on $V'$ is via the mean value type inequality $(\mathbf{H})$).

Lemma 3.8

Assume that $(\mathbf{H})$ holds or $d=1$. Let $x_0\in {\mathbb {R}^{d}}$, $r>0$ and $g(x) = V(\delta _{B(x_0,r)}(x))$. There is a constant $C_5=C_5(d)$ such that

$$\begin{aligned} 0\leqslant \limsup \limits _{t\rightarrow 0}\,\big [-\!\!{\mathcal {A}}_t g(x)\big ]\leqslant \frac{C_5\, H_r}{V(r)} \qquad \text {if }\; {0<}\delta _{B(x_0,r)}(x)< r/4. \end{aligned}$$

(3.9)

Proof

In what follows we use the notation $y=(\tilde{y}, y_d)$, where $y=(y_1,\ldots ,y_d)\in {\mathbb {R}^{d}}$ and $\tilde{y}=(y_1,\ldots ,y_{d-1})$. Without loosing generality we may consider

$$\begin{aligned} x_0=({\tilde{0}},r)\quad \text {and} \quad x=({\tilde{0}}, x_d),\quad \text {where}\; 0<4t<x_d<r/4 \end{aligned}$$

(3.10)

as shown on Fig. 1 (in dimension $d=1$ we mean $y_d=y$, $x_0=r$ and $x_d=x$). We define

$$\begin{aligned} R(y)=V(y_d)-g(y), \quad y\in {\mathbb {R}^{d}}. \end{aligned}$$

We note that $R\geqslant 0$ and $R(x)=0$. Since $V(y_d)$ is harmonic for $X$ in $\{y_d>0\}$, we have

$$\begin{aligned} -\!{\mathcal {A}}_t g(x)&= {\mathcal {A}}_t R(x)= \frac{1}{\mathbb {E}^x\tau _{B(x,t)}}\mathbb {E}^x[R(X_{\tau _{B(x,t)}})]\geqslant 0. \end{aligned}$$

In fact, by (2.29),

$$\begin{aligned} {\mathcal {A}}_t R(x)&= \frac{1}{\mathbb {E}^x\tau _{B(x,t)}}{ \mathbb {E}^x [R(X_{\tau _{B(x,t)}}), X_{\tau _{B(x,t)}}\in B(x,2t)]}\nonumber \\&+\frac{1}{\mathbb {E}^x\tau _{B(x,t)}}\int _{B(x,2t)^c} R(y) \int _{B(x,t)}\nu (y-w)G_{B(x,t)}(x,w)dw dy.\quad \quad \end{aligned}$$

(3.11)

We shall split the integral into several parts. First, if $y\in B_{r/2}^c\subset B(x,t)^c$ and $w\in B(x,t)$, then $\nu (y-w)\leqslant \nu (3y/8)\leqslant \nu (y/4)$, and by (2.27),

$$\begin{aligned} \int _{B(x,t)}\nu (y-w)G_{B(x,t)}(x,w)dw\leqslant \mathbb {E}^x\tau _{B(x,t)}\nu (y/4). \end{aligned}$$

By this, change of variables, subadditivity (2.16) of $V$ and Lemma 3.5,

$$\begin{aligned}&\frac{1}{\mathbb {E}^x\tau _{B(x,t)}}\int _{B_{r/2}^c} R(y) \int _{B(x,t)}\nu (y-w)G_{B(x,t)}(x,w)dw dy\leqslant \int _{B_{r/2}^c}R(y)\nu (y/4)dy\\&\quad \leqslant {\int _{B_{r/2}^c}\nu (y/4)V( {|y|})dy =4^{d+1}\int _{B_{r/8}^c}\nu (y)V(|y|)dy \leqslant c/V(r).} \end{aligned}$$

If $d=1$, then $R(y)=0$ on $B_{r/2}=(-r/2,r/2)$, and the proof is complete.

Fig. 1

The settings for the proofs of Lemma 3.8 (right) and Lemma 3.9 (left)

In what follows we assume that $d\geqslant 2$ and (3.7) holds. We denote (half-ball) $F= B(x_0/2,r/2)\cap \{y_d<r/2\}=\{y: |y|^2/r<y_d<r/2\}$, and we have

$$\begin{aligned} y_d/2\leqslant \delta _{B(x_0,r)}(y)\leqslant y_d \quad \text { and } \quad y_d-\delta _{B(x_0,r)}(y)\leqslant |\tilde{y}|^2/{r}, \quad \text { if } y\in F \end{aligned}$$

(3.12)

(see the right side of Fig. 1). We leave verification of (3.12) to the reader. By (3.7) and (3.12),

$$\begin{aligned} R(y)\leqslant H_rV'(y_d/2) \frac{|\tilde{y}|^2}{r}, \quad y\in F. \end{aligned}$$

(3.13)

If $y\in B(x,2t)\subset F$, then by (3.10) and (3.12) we further have

$$\begin{aligned} R(y)\leqslant 4H_r V'(x_d/4) \frac{t^2}{r}. \end{aligned}$$

(3.14)

By (3.14), (1.4), (2.19) and Lemma 2.5,

$$\begin{aligned}&\!\!\! \frac{1}{\mathbb {E}^x\tau _{B(x,t)}} \mathbb {E}^x [R(X_{\tau _{B(x,t)}}), X_{\tau _{B(x,t)}}\in B(x,2t)]\\&\quad \leqslant \frac{4H_r}{\mathbb {E}^x\tau _{B(x,t)}}\; V'(x_d/4) \frac{t^2}{r} \leqslant c H_rV'(x_d/4) \frac{t^2}{rV^2(t)}\\&\quad \rightarrow {c H_rV'(x_d/4) \frac{\sigma ^2}{r}}, \quad \text {as }t\rightarrow 0. \end{aligned}$$

If $\sigma >0$, then by Lemma 2.2 we have $\sup _{x>0}V'(x)\leqslant 1/\sigma$, hence $V(r)\leqslant r/\sigma$ and so

$$\begin{aligned} V'(x_d/4) \frac{\sigma ^2}{r}\leqslant \frac{\sigma }{r}\leqslant \frac{1}{V(r)}. \end{aligned}$$

If $y\in B_{r/2}\setminus {B(x,2t)}$ and $w\in B(x,t)$, then $|y-w|\geqslant |y-x|/2$. Thus, (3.9) follows if

$$\begin{aligned} \int _{B_{r/2}}R(y)\nu \left( \frac{y-x}{2}\right) dy\leqslant \frac{C_5H_r}{V(r)}. \end{aligned}$$

(3.15)

To prove (3.15), we note the singularity at $y=x\in F$, cover $B_{r/2}$ with sets $\{y\in F:|y_d-x_d|\leqslant |\tilde{y}|\}$, $\{y\in F:|\tilde{y}|<|y_d-x_d| \}$, $\{y\in {\mathbb {R}^{d}}: |\tilde{y}|<r/2, -r/2<y_d\leqslant |y|^2/r \}$, and consider the corresponding integrals. By (3.13), and Lemma 3.7, the first integral does not exceed

$$\begin{aligned} \frac{H_r}{r}\omega _{d{-1}}\int _0^{r}V'(y_d/2)\int _{ |y_d-x_d|}^r \rho ^d\nu (\rho /2)d \rho \, {dy_d}\leqslant \frac{CH_r}{V(r)}. \end{aligned}$$

Similarly, using (3.13) and Lemma 3.7, we bound the second integral by

$$\begin{aligned}&\frac{H_r}{r}\int _0^{r} V'(y_d/2)\nu \left( |y_d-x_d|/2\right) \int _{ |\tilde{y}| < |y_d-x_d|} |\tilde{y}|^2 d\tilde{y}dy_d\\&\quad =\frac{CH_r}{r}\int _0^{r} V'(y_d/2)\nu (|y_d-x_d|/2)|y_d-x_d|^{d+1} dy_d \leqslant \frac{CH_r}{V(r)}. \end{aligned}$$

By a change of variables, subadditivity (2.16) of $V$, and Proposition 3.6, we bound the third integral by

$$\begin{aligned}&\int _0^{r/2}V(s) \int \limits _{rs-s^2\leqslant |\tilde{y}|^2<(r/2)^2}\nu (\tilde{y}/2)d\tilde{y}ds {\leqslant \omega _{d-1}}\int _0^{r/2}\rho ^{d-2}\nu (\rho /2)d\rho \int _0^{{2}\rho ^2/r} V(s) ds\\&\quad \leqslant \frac{2}{r}{\omega _{d-1}}\int _0^{r/2}\rho ^{d}\nu (\rho /2)V({2}\rho ) d\rho \leqslant \frac{C}{V(r)}. \end{aligned}$$

This completes the proof of (3.15), and so the proof of the lemma. $\square$

Lemma 3.9

Assume that $(\mathbf{H})$ holds or $d=1$. Let $x_0\in {\mathbb {R}^{d}}$, $r>0$ and $g(x) = V(\delta _{B^c(x_0,r)}(x))$. There is a constant $C_6=C_6(d)$ such that

$$\begin{aligned} 0\leqslant \limsup _{t\rightarrow 0}{\mathcal {A}}_t g(x)\leqslant \frac{C_6H_r}{V(r)},\qquad \text { if }\quad 0<\delta _{B^c(x_0,r)}(x)< r/4. \end{aligned}$$

(3.16)

Proof

As in the proof of Lemma 3.8, we use the notation $y=(\tilde{y}, y_d)$ and without loosing generality we consider $x=({\tilde{0}}, x_d)$, $0<4t<x_d<r/4$, and $x_0=({\tilde{0}},-r)$ (in dimension $d=1$ we mean $y_d=y$, $x_0={-}r$ and $x_d=x$). This time we define

$$\begin{aligned} R(y)=g(y)-V(y_d), \quad y\in {\mathbb {R}^{d}}. \end{aligned}$$

We have $R\geqslant 0$ and $R(x)=0$. Since $V(y_d)$ is harmonic for $X_t$ at $y_d>0$,

$$\begin{aligned} {\mathcal {A}}_t g(x)&= {\mathcal {A}}_t R(x)= \frac{1}{\mathbb {E}^x\tau _{B(x,t)}}\mathbb {E}^x[R(X_{\tau _{B(x,t)}})]\geqslant 0. \end{aligned}$$

To prove (3.16) we repeat verbatim the proof of Lemma 3.8, starting from (3.11) there, except for the following minor modification: we replace (3.12) with

$$\begin{aligned} y_d\leqslant \delta _{B^c(x_0,r)}(y)\leqslant 3y_d/2 \quad \text { and } \quad \delta _{B^c(x_0,r)}(y)-y_d\leqslant |\tilde{y}|^2/{r}, \quad \text { if } y\in F{,} \end{aligned}$$

(3.17)

where $F= B(x_0/2,r/2)\cap \{y_d<r/2\}$, as before. $\square$

4 Estimates of the expected exit time

Unless explicitly stated otherwise, we keep assuming that $X$ is a unimodal Lévy process in ${\mathbb {R}^{d}}$ with unbounded Lévy–Khintchine exponent $\psi$. The following theorem gives a sharp estimate for the expected exit time of the ball. Recall that the upper bound in Theorem 4.1 actually holds for arbitrary rotation invariant Lévy process, as proved in Lemma 2.3.

Theorem 4.1

If $(\mathbf{H})$ holds, then there is $C_7=C_7(d)$ such that for $r>0$,

$$\begin{aligned} \frac{C_7}{H_r} V(\delta _{B_r}(x))V(r)\leqslant \mathbb {E}^x\tau _{B_r}\leqslant 2 V(\delta _{B_r}(x))V(r), \quad x\in {\mathbb {R}^{d}}. \end{aligned}$$

(4.1)

Proof

Due to Lemma 2.3 it suffices to prove the lower bound in (4.1). Of course it holds on $\overline{B}^c_r$. Denote $s(x)=\mathbb {E}^x \tau _{B_r}$ and $g(x)=V(\delta _{B_r}(x))$, $x\in {\mathbb {R}^{d}}$. By (2.26), domain-monotonicity of the exit times and subadditivity (2.16) of $V$, the bound holds on $\overline{B_{3r/4}}$, i.e. there is $C=C(d)$ so large that

$$\begin{aligned} Cs(x)- V(r)g(x)\geqslant 0 \qquad \text { if } \quad \delta _{B_r}(x)\geqslant r/4. \end{aligned}$$

Let $0<\delta _{B_r}(x)< r/4$. If $t>0$ is small, then by Lemma 3.8 we have $|{\mathcal {A}}_t g(x)|\leqslant C_5H_r/V(r)$, and by (3.2) we obtain

$$\begin{aligned} {\mathcal {A}}_t\left[ (C_5H_r+1) s- V(r)g\right] (x) = -(C_5H_r+1)- V(r) {\mathcal {A}}_t g(x)\leqslant -1. \end{aligned}$$

Let $c=C\vee (C_5H_r+1)$ and $f = c s - V(r)g$, a continuous function. By Corollary 3.1, $f$ cannot attain global minimum on $B_r{\setminus }\overline{B_{3r/4}}$. Since $f\geqslant 0$ elsewhere, $f\geqslant 0$ everywhere. $\square$

The above argument was inspired by the proof of Green function estimates for the ball and stable Lévy processes given by Bogdan and Sztonyk in [13].

Corollary 4.2

If $D$ is bounded, convex and $C^{1,1}$ at scale $r>0$, and if $(\mathbf{H})$ holds, then

$$\begin{aligned} \frac{C_7}{H_r} V(\delta _{D}(x))V(r)\leqslant \mathbb {E}^x\tau _{D}\leqslant V(\delta _{D}(x))V({{\mathrm{diam}}}( D)), \quad x\in {\mathbb {R}^{d}}. \end{aligned}$$

(4.2)

Proof

Fix $x\in D$ and consider a strip $\Pi \supset D$ of width not exceeding ${{\mathrm{diam}}}(D)$ and ball $B\subset D$ of radius $r\vee \delta _D(x)$ such that $\delta _D(x)= \delta _\Pi (x)=\delta _B(x)$. Since $s_B(x)\leqslant s_D(x)\leqslant s_\Pi (x)$, the result follows from (2.18) and Theorem 4.1. $\square$

Remark 4

All the results in this section also hold if $\nu$ is isotropic, infinite and approximately unimodal in the sense of (4.3) below. Here is an example and explanation.

Corollary 4.3

Let $X$ be isotropic with absolutely continuous Lévy measure $\nu (dx)=\nu (|x|)dx$. Let $\nu _0:(0,\infty )\rightarrow (0,\infty )$ be monotone and let $C^*$ be a constant such that

$$\begin{aligned} (C^*)^{-1}\nu _0(r)\leqslant \nu (r)\leqslant C^*\nu _0(r), \quad r>0. \end{aligned}$$

(4.3)

If $(\mathbf{H})$ holds, then there is $c=c(d,C^*)$ such that for $r>0$,

$$\begin{aligned} \mathbb {E}^x\tau _{B_r}\geqslant \frac{c}{H_r} V(\delta _{B_r}(x))V(r), \quad x\in {\mathbb {R}^{d}}.\end{aligned}$$

Proof

Let $Y$ be unimodal with characteristic function $\psi ^Y(\xi )=\sigma {|\xi |^2}+\int _{{\mathbb {R}^{d}}}(1-\cos \langle \xi , z\rangle )\nu _0(|z|)dz$. By Proposition 2.4 we have $V^Y(r)\approx V(r)$, $r>0$. By Proposition 3.6,

$$\begin{aligned} \int ^r_0V(\rho )\rho ^d\nu (\rho )d\rho \approx \int ^r_0V^Y(\rho )\rho ^d\nu _0(\rho )d\rho \leqslant C\frac{r}{V(r)},\quad r>0. \end{aligned}$$

(4.4)

The inequality and approximate monotonicity of $\nu$ yield extensions of Lemmas 3.7 and 3.8, from which the present corollary follows in a similar manner as Theorem 4.1. $\square$

For $r>0$ we define functions

$$\begin{aligned} \mathcal {I}(r)=\inf _{ 0<\rho \leqslant r/2}\left[ \nu (B_{r}{\setminus }B_\rho ) V^2(\rho )\right] , \end{aligned}$$

(4.5)

and

$$\begin{aligned} \mathcal {J}(r)=\inf _{ 0<\rho \leqslant r}\left[ L(\rho ) V^2(\rho )\right] . \end{aligned}$$

(4.6)

We note that $\mathcal {J}$ is non increasing. By (2.23),

$$\begin{aligned} 0\leqslant \mathcal {I}(2r)\leqslant \mathcal {J}(r)\leqslant c(d),\qquad r>0. \end{aligned}$$

(4.7)

We shall use $\mathcal {J}$ immediately, but $\mathcal {I}$ shall only be discussed and used later, in Sects. 5 and 6.

Lemma 4.4

Let $(\mathbf{H})$ hold. Denote $D=B_{1}^c$. Let $0<r<1$, $x\in D$ and $0<\delta _D(x)\leqslant r/2$. Let $x_0=x/|x|$ and $D_1= B(x_0, r)\cap D$. There is $C_{8}=C_8(d)$ such that

$$\begin{aligned} \mathbb {E}^x \tau _{D_1}\leqslant C_{8}\frac{H_1 }{\left( \mathcal {J}(1)\right) ^{2}} V(\delta _D(x))\,V(r). \end{aligned}$$

(4.8)

Proof

We may and do assume that $x=(\tilde{0}, x_d)$ with $1<x_d\leqslant 1+r/2$. If $a,b,c\geqslant 0$, $k\geqslant 2$, $a-b+c\geqslant 0$ and $b\geqslant kc$, then $a\geqslant b-c\geqslant (k-1)c\geqslant kc/2$, so $c\leqslant 2a/k$. We shall use this observation to compare $a({v}) = {V(\delta _{D}(v))}$, $b(v)= \mathbb {E}^v a(X_{\tau _{D_1}})$ and $s(v)= \mathbb {E}^v \tau _{D_1}$, where $v\in {\mathbb {R}^{d}}$. We first let $0<r\leqslant 1/4$, and consider

$$\begin{aligned} f(v)= a(v) -b(v)+\frac{{C_6H_1+1}}{V(1)} s(v),\quad v\in {\mathbb {R}^{d}}. \end{aligned}$$

If $v\notin D_1$ then $a(v) = b(v)$, $s(v)=0$, and so $f(v)= 0$. By Lemmas 2.9,  3.4 and 3.5, and by subadditivity of $V$, $f\in C_0(D_1)$. If $v\in D_1$, and $t>0$ is small enough, then Lemma 3.9 and (3.2) yield

$$\begin{aligned} {\mathcal {A}}_t f(v) \leqslant \frac{{C_6H_1}}{V(1)}-\frac{{C_6H_1+1}}{V(1)}<0. \end{aligned}$$

By Lemma 3.2, $f\geqslant 0$ on $D_1$. Consider a point $z^*\in \partial B_1\cap \partial B(x_0,r)$. We have $z^*_d=1-r^2/2$. Let $F=\{y: y_d>1+r\}$. For all $z\in D_1$, $F-z= \{y: y_d>1+r-z_d\}\supset \{y: y_d>r+r^2/2\}= F-z^*$, hence $\inf _{z\in D_1}\nu (F-z)= \nu (F-z^*)= (1/2)L_1(r+r^2/2)$. By Ikeda–Watanabe,

$$\begin{aligned} b(v)&\geqslant V(r) {\mathbb {P}}^v( X_{\tau _{D_1}}\in F)\geqslant V(r) {\mathbb {E}}^v \tau _{D_1} \inf _{z\in D_1}\nu (F-z)\\&= {\frac{1}{2}} V(r) \mathbb {E}^v \tau _{D_1} L_1(r+r^2/2)\geqslant {\frac{1}{2}}V(r) \mathbb {E}^v \tau _{D_1} L_1(9r/8). \end{aligned}$$

Since $X_t$ is rotation invariant, there is $c_1=c_1(d)$ such that $L_1(9r/8)\geqslant {8}c_1L(2r)$. Indeed, $B_{2r}^c$ may be covered by a finite number of rotations of $\{y\in {\mathbb {R}^{d}}: |y_d|>9r/8\}$. Thus, for $r\leqslant 1/4$, we have $L_1(9r/8)\geqslant {8}c_1L(2r)\geqslant \frac{ {8}c_1\mathcal {J}(1)}{V^2(2r)} \geqslant \frac{{2}c_1\mathcal {J}(1)}{V^2(r)}$, hence

$$\begin{aligned} b(v)\geqslant \mathbb {E}^v \tau _{D_1} \frac{c_1\mathcal {J}(1)}{V(r)}{=}\frac{c_1\mathcal {J}(1)}{C_6H_1+1}\frac{V(1)}{V(r)}\frac{C_6H_{1}+1}{V(1)} s(v). \end{aligned}$$

If $\frac{c_1\mathcal {J}(1)}{C_6H_1+1}\frac{V(1)}{V({1/4})}\geqslant 2$, then we let $r_0=1/4$, else we pick $r_0>0$ so that $\frac{c_1\mathcal {J}(1)}{C_6H_1+1}\frac{V(1)}{V({r_0})}= 2$. By the observation at the beginning of the proof, for $0<r\leqslant r_0$, we have $s(v)\leqslant 2V(\delta _D(v))V(r)/(c_1\mathcal {J}(1))$ for all $v$, in particular for $v=x$.

For $r_0<r<1$ we proceed in the following standard way. First assume that $\delta _{D}(x)\leqslant r_0/2$, and let $D^\prime = B(x_0, r_0)\cap D$. Then by the strong Markov property,

$$\begin{aligned} s(x)=\mathbb {E}^x \tau _{D_1}=\mathbb {E}^x \tau _{D^\prime }+\mathbb {E}^xs(X_{\tau _{D^\prime }}). \end{aligned}$$

As stated in Theorem 4.1, $s(x)\leqslant 2V^2(r)$. By Lemma 2.7, we thus obtain,

$$\begin{aligned} \mathbb {E}^x s(X_{\tau _{D^\prime }})&\leqslant 2V^2(r)\mathbb {P}^x(|X_{\tau _{D^\prime }}-x_0|\geqslant r_0)\\&\leqslant 2C_1 V^2(r)\frac{\mathbb {E}^x \tau _{D^\prime }}{V^2(r_0)}. \end{aligned}$$

If this is combined with the estimates already proved, then $c_2=c_2(d)$ exists such that

$$\begin{aligned} s(x)&\leqslant (2C_1+1) \mathbb {E}^x \tau _{D^\prime } \frac{V^2(r)}{V^2(r_0)}\\&\leqslant c_2 \frac{V(r)}{\mathcal {J}(1)V(r_0)} V(\delta _D(x))V(r)\leqslant c_2 \frac{V(1)}{\mathcal {J}(1)V(r_0)} V(\delta _D(x))V(r). \end{aligned}$$

If $\delta _D(x)\geqslant r_0/2$, then by Lemma 2.3 and subadditivity of $V$, we trivially have

$$\begin{aligned} s(x)\leqslant 2V^2(r)\leqslant \frac{2V(r)}{V(r_0/2)}V(\delta _D(x))V(r)\leqslant \frac{4V(1)}{V(r_0)}V(\delta _D(x))V(r). \end{aligned}$$

Summarizing, by taking $c_3=4+ c_2$, in all the cases we get

$$\begin{aligned} \mathbb {E}^x \tau _{D_1}\leqslant c_3 \frac{V(1)}{V(r_0)} \left( 1+ \frac{1}{\mathcal {J}(1)}\right) V(\delta _D(x))V(r). \end{aligned}$$

By the choice of $r_0$, subadditivity of $V$ and (4.7), $V(1)/V(r_0)\leqslant 4+ {2}(C_6H_1+1)/(c_1\mathcal {J}(1))\leqslant c_4H_1/\mathcal {J}(1)$, where $c_4=c_4(d)$. Therefore,

$$\begin{aligned}\mathbb {E}^x \tau _{D_1}\leqslant c_4H_1 \frac{\mathcal {J}(1)+1}{\mathcal {J}(1)^2 } V(\delta _D(x))V(r). \end{aligned}$$

This is equivalent to (4.8). $\square$

Corollary 4.5

Let $(\mathbf{H})$ hold. Denote $D=B_{R}^c$. Let $0<r<R$, $x\in D$, $0<\delta _D(x)\leqslant r/2$, $x_0=xR/|x|$. If $D_1= B(x_0, r)\cap D$, then

$$\begin{aligned} \mathbb {E}^x \tau _{D_1}\leqslant C_{8}\frac{H_R}{\left( \mathcal {J}(R)\right) ^{2}} V(\delta _D(x))\,V(r). \end{aligned}$$

(4.9)

Proof

Let $R>0$, $Y_t=X_{ t}/R$ and denote by $V_Y,\, \tau ^Y_{B},\, L_Y,\, \mathcal {J}^Y_\infty , {H^Y_\cdot }$ the quantities $V$, $\tau _B$, $L$, $\mathcal {J}, {H_\cdot }$ corresponding to $Y$. By (2.14) and (2.17), we infer that $V_Y(s)= V(Rs),\, s\geqslant 0$. Furthermore, $L_Y(s)= L(Rs)$ for $s>0$. Hence, we obtain $V^{2}_Y(s)L_Y(s )= V^2(Rs)L(Rs),$ which shows that $\mathcal {J}^Y(1) =\mathcal {J}(R)$. Also, ${H^Y_1}=H_R$ and

$$\begin{aligned} \mathbb {E}^x\tau _{D_1}=\mathbb {E}^{x/R}\tau ^Y_{D_1/R}. \end{aligned}$$

Here the expectation on the right hand side corresponds to $Y$. Lemma 4.4 finishes the proof. $\square$

The above argument shall be called scaling (a different, weak scaling is discussed in Sect. 5).

The following is one of our main results.

Theorem 4.6

If $(\mathbf{H})$ holds and $D\subset {\mathbb {R}^{d}}$ is open, bounded and $C^{1,1}$ at scale $r>0$, then $C_9=C_9(d)$ and $C_{10}=C_{10}(d)$ exist such that

$$\begin{aligned} \frac{C_9}{H_r}\, {V(\delta _{D}(x))V(r)}\leqslant \mathbb {E}^x\tau _{D}\leqslant C_{10}\frac{H_r}{\left( \mathcal {J}(r)\right) ^{2}} \, \frac{V^2({{\mathrm{diam}}}D)}{V^2(r)}\, {V(\delta _{D}(x))V(r)}, \quad x\in {\mathbb {R}^{d}}. \end{aligned}$$

Proof

Denote $s(x)={\mathbb {E}}^x\tau _D$. By Lemma 2.3, $s(x)\leqslant {2} V^2({{\mathrm{diam}}}( D))$. Let $Q\in \partial D$ be such that $|x-Q|=\delta _D(x)$. Let $\delta _D(x)\leqslant r/2$. Since $D$ is $C^{1,1}$ at scale $r$, there exist $x_1\in D^c$ and $x_2\in D$ such that $B(x_1,r)\subset D^c$, $B(x_2,r)\subset D$ and $\{Q\}= \overline{B(x_1,r)}\cap \overline{B(x_2,r)}$. Let $D_1=B(Q,r)\cap D$. By the strong Markov property and (2.25),

$$\begin{aligned} s(x)&= \mathbb {E}^x\tau _{D_1}+\mathbb {E}^xs(X_{\tau _{D_1}})\leqslant \mathbb {E}^x\tau _{D_1}+{2}V^2({{\mathrm{diam}}}(D))\mathbb {P}^x(|X_{\tau _{D_1}}-Q|>r)\\&\leqslant \mathbb {E}^x\tau _{D_1}\left( 1+{2}C_1\frac{V^2({{\mathrm{diam}}}(D))}{V^2(r)}\right) . \end{aligned}$$

Corollary 4.5 yields the upper bound, since $\mathbb {E}^x\tau _{D_1}\leqslant \mathbb {E}^x\tau _{D_2}$, where $D_2= B(Q,r)\cap \overline{B(x_1,r)}^c$. The lower bound is a consequence of Theorem 4.1, because $s(x)\geqslant \mathbb {E}^x\tau _{{B(x_2,r)}}$.

For the case $\delta _D(x)\geqslant r/2$, we see from (2.26) that $s(x)\geqslant \mathbb {E}^x\tau _{{B(x,\delta _D(x))}}\geqslant C_1^{-1}V^2(\delta _D(x))\geqslant (2C_1)^{-1}V(\delta _D(x))V(r)$. By this, the general upper bound $s(x)\leqslant 2V^2({{\mathrm{diam}}}( D))$ and the observations that $H_r\geqslant 1$ and $\mathcal {J}(r)\leqslant c(d)$, we finish the proof. $\square$

For instance, $V^2({{\mathrm{diam}}}D)/V^2(r)$ is bounded by the square of the distortion of $D$, if $r$ equals the localization radius of $D$.

In the one-dimensional case in the proof of Theorem 4.6 we may apply (2.18) instead of Theorem 4.1 and Corollary 4.5, to obtain the following improvement.

Corollary 4.7

If $X$ is a symmetric Lévy process in $\mathbb {R}$ with unbounded Lévy-Khintchine exponent, and $D\subset \mathbb {R}$ is open, bounded and $C^{1,1}$ at scale $r>0$, then absolute constant $c\geqslant 1$ exists such that $c^{-1}{V(\delta _{D}(x))V(r)} \leqslant { \mathbb {E}^x\tau _{D}}\leqslant c V^2({{\mathrm{diam}}}D)V^{-2}(r)\,{V(\delta _{D}(x))V(r)}$ for $x\in \mathbb {R}$.

5 Scaling and its consequences

Let $X$ be an isotropic unimodal Lévy process in ${\mathbb {R}^{d}}$ with infinite Lévy measure $\nu$. In view of the literature of the subject (cf. [10, 34, 35, 39]), power-like asymptotics of the characteristic exponent of $X$ is a natural condition to consider. Let $I=({\underline{\theta }},\infty )$, where ${\underline{\theta }}\in [0,\infty )$ and let $\phi \geqslant 0$ be a non-zero function on $(0,\infty )$. We say that $\phi$ satisfies the weak lower scaling condition (at infinity) if there are numbers ${\underline{\alpha }}>0$ and ${\underline{c}}{\in (0,1]}$, such that

$$\begin{aligned} \phi (\lambda \theta )\geqslant {\underline{c}}\lambda ^{\,{\underline{\alpha }}} \phi (\theta )\quad \hbox {for}\quad \lambda \geqslant 1, \quad \theta {\in I}. \end{aligned}$$

(5.1)

In short we say that $\phi$ satisfies WLSC(${\underline{\alpha }}, {\underline{\theta }},{{\underline{c}}}$) and write $\phi \in \mathrm WLSC ({\underline{\alpha }}, {\underline{\theta }},{\underline{c}})$. If $\phi \in \mathrm WLSC ({\underline{\alpha }},0,{\underline{c}})$, then we say that $\phi$ satisfies the global weak lower scaling condition.

Similarly, we consider $I=({\overline{\theta }},\infty )$, where ${\overline{\theta }}\in [0,\infty )$ and we say that the weak upper scaling condition holds if there are numbers ${\overline{\alpha }}{<2}$ and ${\overline{C}}{\in [1,\infty )}$ such that

$$\begin{aligned} \phi (\lambda \theta )\leqslant {\overline{C}}\lambda ^{\,{\overline{\alpha }}} \phi (\theta )\quad \hbox {for}\quad \lambda \geqslant 1, \quad \theta {\in I}. \end{aligned}$$

(5.2)

In short, $\phi \in \mathrm WUSC ({\overline{\alpha }}, {\overline{\theta }},{\overline{C}})$. For global weak upper scaling we require ${\overline{\theta }}=0$ in (5.2). We write $\phi \in$ WLSC or WUSC if the actual values of the parameters are not important. We shall study consequences of WUSC and WLSC for the characteristic exponent $\psi$ of $X_t$.

Recall that $\psi$ is a radial function and we use the notation $\psi (u)=\psi (x)$, where $x\in {\mathbb {R}^{d}}$ and $u=|x|$. Our estimates below are expressed in terms of $V$, $\psi$ or $\psi ^*$. In view of Proposition (2.4), these functions yield equivalent descriptions ($\psi$ or $\psi ^*$ are even comparable, see (3.1)). Our main goal is to find connections between the scaling conditions on $\psi$ and the magnitude of the quantities $\mathcal {J}$ and $\mathcal {I}$ defined in (4.6) and (4.5). In the preceding section we saw that $\mathcal {J}$ plays a role in estimating the expected exit time from $C^{1,1}$ open sets. The next three results prepare analysis of survival probabilities in Sect. 6. The first one comes from [10, Corollary 15].

Lemma 5.1

$C=C(d)$ exists such that if $\psi {\in }$ WUSC$({\overline{\alpha }},{\overline{\theta }},{\overline{C}})$, $a=[(2-{\overline{\alpha }})C]^{\frac{2}{2-{\overline{\alpha }}}}{\overline{C}}^{\frac{{\overline{\alpha }}-2}{2}}$, then

$$\begin{aligned} L(r)\geqslant a \psi (r^{-1}), \qquad {0<}r\leqslant \sqrt{a}/{\overline{\theta }}. \end{aligned}$$

The following result makes use of the complete Bernstein function $\phi (\lambda )\approx \psi (\sqrt{\lambda })$, constructed in the proof of [10, Theorem 26]. For the convenience of the reader we repeat some of the arguments from [10].

Proposition 5.2

(i) $\psi$ satisfies WUSC if and only if there is $R >0$, such that $\mathcal {J}(R )>0$. (ii) $\psi$ satisfies WUSC and WLSC (global WUSC and WLSC) if and only if for some $R >0$ ($R=\infty$, resp.) we have $\inf _{ {r} < R }\mathcal {I}(r)>0$.

Proof

Assume that $\psi$ satisfies WUSC$(\beta _1,{{\overline{\theta }}},{\overline{C}})$. By Lemma 5.1 and (2.19), there is a constant $c_1$ such that $L(r)V^2(r)\geqslant c_1>0$ for $r\leqslant \sqrt{a}/{{\overline{\theta }}}$, and so $\mathcal {J}(r)\geqslant c_1>0$ for such $r$. On the other hand, if $R\in (0,\infty )$ is given and $\mathcal {J}(R ) >0$, then

$$\begin{aligned} L(r)&\geqslant \mathcal {J}(R )/V^2(r), \qquad r\leqslant R. \end{aligned}$$

(5.3)

By [44, proof of Theorem 6.2] and (2.3), the following defines a complete Bernstein function:

$$\begin{aligned} \varphi (\lambda )=\int ^{\infty }_{0}\frac{\lambda }{\lambda +s}s^{-1}\nu (s^{-1/2})s^{-d/2}ds=\int ^{\infty }_{0}\left( 1-e^{-\lambda u}\right) \mu (u)du,\quad \lambda \geqslant 0, \end{aligned}$$

where $\mu (u)=\mathcal {L}[\nu (s^{-1/2})s^{-d/2}](u)$. In fact, by changing variables, and (2.5) for $\lambda >0$ we have

$$\begin{aligned} \varphi (\lambda )&=2\int ^{\infty }_{0}\frac{\lambda u^2}{\lambda u^2+1}\nu (u)u^{d-1}du\approx \int ^{\infty }_{0}\left[ 1\wedge (\lambda u^2)\right] \nu (u)u^{d-1}du=\omega _d^{-1}h\left( \lambda ^{-1/2}\right) . \end{aligned}$$

By (2.19), there exists $c_1=c_1(d)$ such that

$$\begin{aligned} c_1 \varphi ( \lambda ) \leqslant \psi \left( \sqrt{\lambda } \right) \leqslant c_1^{-1} \varphi (\lambda ), \quad \lambda \geqslant 0. \end{aligned}$$

(5.4)

Since $\varphi$ is a complete Bernstein function, $\varphi _1(\lambda )=\lambda /\varphi (\lambda )$ is a special Bernstein function (see [44, Definition 11.1 and Proposition 7.1]). Since $X_t$ is pure-jump, $\lim _{|\xi |\rightarrow \infty } \psi ({\xi })/|\xi |^2=0$. Thus, $\lim _{\lambda \rightarrow \infty }\varphi _1(\lambda )=\infty$. By [44, (11.9) and Theorem 11.3], the potential measure of the subordinator with the Laplace exponent $\varphi _1$ is absolutely continuous with the density function $f(s)= \int _s^\infty \mu (u)du$ since $\varphi (0)=0$. In particular $\mathcal {L}f=1/\varphi _1$ (see [44, (5.20)]). We obtain

$$\begin{aligned} \nu (x)\leqslant \frac{\int ^\infty _{|x|^{-2}}e^{-s |x|^{2}}\nu (s^{-1/2})s^{-d/2}ds}{\int ^\infty _{|x|^{-2}}e^{-s |x|^{2}}s^{-d/2}ds}\leqslant \frac{\mu (|x|^2)}{\Gamma (1-d/2,1)|x|^{d-2}} , \quad x\ne 0, \end{aligned}$$

(5.5)

where $\Gamma (1-d/2,1)=\int _1^\infty e^{-u}u^{-d/2}du$ is the upper incomplete gamma integral. Hence,

$$\begin{aligned} f(r)\geqslant c \int _r^\infty u^{(d-2)/2}\nu (u^{1/2})du =cL(r^{1/2}), \quad r>0. \end{aligned}$$

(5.6)

By (5.5), (5.3), (2.19) and (5.4)

$$\begin{aligned} f(r)&\geqslant c{\mathcal {J}(R )}/V^2(r^{1/2})\geqslant c_2 \varphi (r^{-1})=\frac{c_2}{ {r}\varphi _1(r^{-1})},\qquad 0<r\leqslant R^2. \end{aligned}$$

(5.7)

Since $f$ is decreasing and $\mathcal {L}f(u)=1/\varphi _1(u)$, by [10, Lemma 5] we obtain

$$\begin{aligned} f(s)\leqslant \frac{1}{\gamma (2,1)s^2}\left( \frac{1}{\varphi _1}\right) '(s^{-1}) =\frac{1}{\gamma (2,1)s^2}\frac{\varphi _1'(s^{-1})}{\varphi _1^2(s^{-1})},\qquad s>0, \end{aligned}$$

where $\gamma (2,1)=\int _0^1 e^{-u}u \ du$ is the lower incomplete gamma integral. By (5.7), $c_3\varphi _1(\lambda )\leqslant \lambda \varphi '_1(\lambda )$, where $\lambda > 1/ R^2$ and $c_3=c_2 \gamma (2,1)$. It follows that $\lambda ^{-c_3}\phi _1(\lambda )$ is nondecreasing on $( R^{-2},\infty )$. By [10, Lemma 11], $\varphi _1\in \mathrm WLSC (c_3, {R}^ {-2},1)$. Since $\varphi _1$ is concave, $\lambda \varphi _1'(\lambda )\leqslant \varphi _1(\lambda )$, hence $c_3\leqslant 1$. In fact, $\varphi _1$ is not a linear function, because $\varphi$ is unbounded, and so $c_3<1$. Therefore $\varphi \in \mathrm WUSC (1-c_3,{R}^ {-2},1)$, and so $\psi \in \mathrm WUSC (2(1- {c_3}),R^{-1},c_1^{-2})$, by (5.4).

To prove the second part of the statement we suppose that $\psi$ satisfies WUSC$(\beta _1,\theta ,{\overline{C}})$ and WLSC$(\beta _2,\theta ,{\underline{c}})$. By [10, Corollary 22] and (2.19),

$$\begin{aligned} \nu (x)\geqslant \frac{c^*}{ V^2(|x|)|x|^d},\quad |x|\leqslant b/\theta . \end{aligned}$$

If $2 \rho \leqslant b/\theta$, then by monotonicity of $V$ we have

$$\begin{aligned} V^2( \rho )\nu (B_{2\rho }{\setminus } B_{\rho })\geqslant V^2(\rho )\int _{B_{2\rho }\setminus B_{\rho }} \frac{c^*dx}{ V^2(|x|)|x|^d}\geqslant \int _{B_{\rho }\setminus B_{\rho /2}} c^*\frac{dx}{ |x|^d}. \end{aligned}$$

Therefore $\mathcal {I}(r)\geqslant c^*(1-(1/2)^d)\omega _d/d$ for all $r\leqslant b/\theta$, as needed.

To prove the reverse implication we assume that there exist constants $c^*$ and $R$, such that for $0< r< R$, $\mathcal {I}( r)\geqslant c^*$. By radial monotonicity of $\nu$,

$$\begin{aligned} \nu (x)\geqslant \frac{c^*}{|B_ 2|-|B_{1}|}\frac{1}{V^2(|x|)|x|^d}, \quad |x|< R /2. \end{aligned}$$

By (2.19) and [10, Theorem 26], we obtain WLSC and WUSC for $\psi$. $\square$

Proposition 5.3

If $\psi$ satisfies WUSC but not WLSC, then $\liminf _{r\rightarrow 0}\mathcal {I}(r)=0$ but there is ${R} >0$ such that $\mathcal {I}({r} )>0$ for $r< R$.

Proof

Let $R=2\sup \{r: \nu (r)>0\}$. We have $R>0$. If $\psi$ satisfies WUSC, then by Lemma 5.1 and Proposition 2.4, there are $c_1,r_1>0$, such that $L(\rho )\geqslant c_1/V^2(\rho )$ for $\rho < r_1$. Since $\lim _{\rho \rightarrow 0}V(\rho )=0$,

$$\begin{aligned} \liminf _{\rho \rightarrow 0}V^2(\rho )\nu (B_r{\setminus } B_\rho )=\liminf _{\rho \rightarrow 0}V^2(\rho )L(\rho )\geqslant c_1, \end{aligned}$$

for every $r>0$. Fix $r\in (0,R)$. There is $r_2>0$ such that

$$\begin{aligned} V^2(\rho )\nu (B_r{\setminus } B_\rho )\geqslant c_1/2 \quad \text { if } \rho \leqslant r_2. \end{aligned}$$

If $r_2< \rho \leqslant r/2$, then by monotonicity of $V$,

$$\begin{aligned} V^2(\rho )\nu (B_r{\setminus }B_\rho )\geqslant V^2(r_2)\nu (B_r{\setminus } B_{r/2})>0, \end{aligned}$$

hence $\mathcal {I}(r)>0$. If $\liminf _{r\rightarrow 0}\mathcal {I}(r)>0$, then by Proposition 5.2, $\psi$ satisfies also WLSC. $\square$

5.1 Hitting a ball

We shall estimate the probability that $X$ ever hits a fixed ball of radius $R>0$. If $X$ is transient and its starting point is far from the ball, then the probability of such an event is small; $X$ instead drifts to infinity with probability bounded below by a positive constant. Indeed, define

$$\begin{aligned} U(x)=\int ^{\infty }_{0}p_t(x)dt,\qquad x\in {\mathbb {R}^{d}}, \end{aligned}$$

the potential kernel of $X$. If the process is transient [40], then $U$ is finite almost everywhere, in fact on ${\mathbb {R}^{d}{\setminus }\{0\}}$. This is the case, e.g. if $d\geqslant 3$. We denote by ${{\mathrm{{ {Cap}}}}}$ the capacity with respect to $X$. Recall that for every non-empty compact set $A\subset {\mathbb {R}^{d}}$ there exists a measure $\mu _A$, supported on $A$ (see, e.g., [6, Section II.2]), called the equilibrium measure, such that

$$\begin{aligned} U\mu _A(x)=\int U(x-y)\mu _A(dy)=\mathbb {P}^x(\tau _{A^c}<\infty ),\quad x\in {\mathbb {R}^{d}}, \end{aligned}$$

(5.8)

and $\mu _A(A)={{\mathrm{{ {Cap}}}}}(A)$. The following two lemmas were proved in [24].

Lemma 5.4

(Theorem 3 of [24]) If $d\geqslant 3$, then there is $C_{15}=C_{15}(d)$ such that

$$\begin{aligned} U(x)\leqslant C_{15}\frac{V^2(|x|)}{|x|^d}, \qquad x\in \mathbb {R}^d. \end{aligned}$$

We note in passing that lower bounds for $U$ are given in [24] under WLSC.

Lemma 5.5

(Proposition 3 of [24]) If $d\geqslant 3$, then there is $C_{16}=C_{16}(d)$ such that

$$\begin{aligned} C_{16}^{-1}\frac{ R^d}{V^2(R)} \leqslant {{\mathrm{{ {Cap}}}}}(\overline{B_R})\leqslant C_{16}\frac{ R^d}{V^2(R)}, \qquad R>0. \end{aligned}$$

If $\psi \in$ WUSC(${\overline{\alpha }},0,{\overline{C}}$) and $d>{\overline{\alpha }}>0$, then the process $X$ is transient (even if $d<3$), and we may extend the two previous lemmas by using the weak upper scaling condition.

Lemma 5.6

If $\psi \in \mathrm WUSC ({\overline{\alpha }},0,{\overline{C}})$ and ${\overline{\alpha }}<d\leqslant 2$, then $c=c(d,{\overline{\alpha }},{\overline{C}})$ exists such that

$$\begin{aligned} U(x)\leqslant c\frac{V^2(|x|)}{|x|^d}, \qquad x\in \mathbb {R}^d. \end{aligned}$$

Proof

Using the global upper weak scaling condition instead of [24, Lemma 1] one can prove as in [24, Lemma 6] that for $d>{\overline{\alpha }}$,

$$\begin{aligned} \mathcal {L}\left( \int _{B_{\sqrt{\cdot }}}U(y)dy\right) (\lambda )\approx \frac{1}{\lambda \psi ^{*}(\sqrt{\lambda })}, \end{aligned}$$

(5.9)

and then we proceed as in the proof of the first part of [24, Theorem 3]. $\square$

Lemma 5.7

If $\psi \in \mathrm WUSC ({\overline{\alpha }},0,{\overline{C}})$ and ${\overline{\alpha }}<d\leqslant 2$, then $c=c(d,{\overline{\alpha }},{\overline{C}})$ exists such that

$$\begin{aligned} c^{-1}\frac{r^d}{V^2( r)} \leqslant {{\mathrm{{ {Cap}}}}}(\overline{B_r})\leqslant c\frac{r^d}{V^2(r)}, \quad r>0. \end{aligned}$$

Proof

We follow the proof of [24, Proposition 3], using (5.9) instead of [24, Lemma 6]. $\square$

As a consequence of the above lemmas we obtain the following upper bound of the probability that the process ever hits a ball of arbitrary radius, a close analogue of a well known Brownian result. We note that [39, Lemma 2.5] gives the inequality (5.10), in fact comparability of both sides of (5.10), for $d\geqslant 3$ under the global weak lower scaling condition.

Proposition 5.8

For $d\geqslant 3$ there exists a constant $C_{17}=C_{17}(d)$ such that for $|x|>R>0$,

$$\begin{aligned} \left. \mathbb {P}^x(\tau _{\overline{B}^c_R}<\infty )\leqslant C_{17} \frac{ V^2(|x|)}{|x|^d}\Big /\frac{ V^2(R)}{R^d}\right. . \end{aligned}$$

(5.10)

If ${\overline{\alpha }}<d\leqslant 2$ and $\psi \in$WUSC(${\overline{\alpha }},0,{\overline{C}}$), then (5.10) holds with $C_{17}=C_{17}(d,{\overline{\alpha }},{\overline{C}})$.

Proof

We have

$$\begin{aligned} \mathbb {P}^x(\tau _{\overline{B}^c_R}<\infty )=\int _{\overline{B_R}} U(y-x)\mu _{B_R}(dy). \end{aligned}$$

By Lemma 5.4, for $y\in B_R$ and $|x|\geqslant 2R$ we get

$$\begin{aligned} U(x-y)\leqslant 2^dC_{15}|x|^{-d}V^2(|x|). \end{aligned}$$

Hence, by Lemma 5.5,

$$\begin{aligned} \mathbb {P}^x(\tau _{\overline{B}^c_R}<\infty )\leqslant 2^dC_{15}|x|^{-d}V^2(|x|){{\mathrm{{ {Cap}}}}}(\overline{B_R})\leqslant 2^dC_{15}C_{16} \frac{R^{d}V^2(|x|)}{|x|^{d}V^2(R)}. \end{aligned}$$

Since $[R^{d}V^2(|x|)]/[|x|^{d}V^2(R)]\geqslant 2^{-d}$, for $|x|\leqslant 2R$ we have

$$\begin{aligned} \mathbb {P}^x(\tau _{\overline{B}^c_R}<\infty )\leqslant 2^d(C_{15}C_{16}+1) \frac{R^{d}V^2(|x|)}{|x|^{d}V^2(R)}, \quad |x|>R. \end{aligned}$$

To prove the second claim we use Lemma 5.6 and 5.7 above instead of 5.4 and 5.5. $\square$

The following result is important in Sect. 6.

Corollary 5.9

If $d\geqslant 3$, then $c=c(d)$ exists such that

$$\begin{aligned} \mathbb {P}^x(\tau _{\overline{B}^c_R}=\infty )\geqslant 1/2, \qquad |x|\geqslant c R. \end{aligned}$$

If ${\overline{\alpha }}<d\leqslant 2$ and $\psi \in$WUSC(${\overline{\alpha }},0,{\overline{C}}$), then the above inequality holds with $c=c(d,{\overline{\alpha }},{\overline{C}})$.

6 Estimates of survival probability

In this section we assume that $X$ is a pure-jump (isotropic) unimodal Lévy process with infinite Lévy measure.

Proposition 6.1

Let $(\mathbf{H})$ hold. There are $C_{11}=C_{11}(d)<1$ and $C_{12}=C_{12}(d)$ such that if $R> 0$ and $t\leqslant C_{11} V^2(R)$, then

$$\begin{aligned} \mathbb {P}^x(\tau _{B_{R}}>t)&\geqslant C_{12}\,\frac{\mathcal {I}(R)}{H_R}\left( \frac{V(\delta _{B_{R}}(x))}{\sqrt{t}}\wedge 1\right) . \end{aligned}$$

Proof

Let $R=1$ and $C_{11}=C_3/64$. Due to Corollary 2.8 and subadditivity of $V$,

$$\begin{aligned} {\mathbb {P}^0}(\tau _{B_{r/8}}>C_{11} V^2(r))\geqslant 1/2. \end{aligned}$$

(6.1)

Suppose that $0<t\leqslant C_{11} V^2(1)$ and pick $r\leqslant 1$ such that $t = C_{11} V^2(r).$ Let $x\in B_{1}$. If $\delta _{B_{1}}(x)\geqslant {r/8},$ then $\mathbb {P}^x(\tau _{B_{1}}>t)\geqslant 1/2$ by (6.1). To complete the proof for $R=1$, it is enough to consider the case $\delta _{B_{1}}(x)< {r/8}$. Let $\delta _{B_{1}}(x)<{r/8}$. Let $r_0=r/2\wedge 1/4$ and $D_r=B_{1}{\setminus }B_{1-r_0}$. Notice that $B(z,r/4)\subset B_1$ for $z\in B_{1-r_0}$. By the strong Markov property,

$$\begin{aligned} \mathbb {P}^x(\tau _{B_{1}}>t)&\geqslant \mathbb {E}^x\left[ \mathbb {P}^{X_{\tau _{D_r}}}(\tau _{B_{1}}>t); X_{\tau _{D_r}}\in B_{1-r_0}\right] \\&\geqslant \inf _{z\in B_{1-r_0}}\mathbb {P}^z(\tau _{B_{1}}>t) \mathbb {P}^x\left[ X_{\tau _{D_r}}\in B_{1-r_0}\right] \\&\geqslant \mathbb {P}^0(\tau _{B_{r/4}}>C_{11}V^{2}(r)) \mathbb {P}^x\left[ X_{\tau _{D_r}}\in B_{1-r_0}\right] \\&\geqslant (1/2) \mathbb {P}^x\left[ X_{\tau _{D_r}}\in B_{1-r_0}\right] . \end{aligned}$$

If $|z_0|={1}$, then by the Ikeda–Watanabe formula, isotropy and monotonicity of the Lévy density,

$$\begin{aligned} \mathbb {P}^x\left[ X_{\tau _{D_r}}\in B_{1-r_0}\right] \geqslant \mathbb {E}^x \tau _{D_r}\inf _{z\in D_r}\nu (z-B_{1-r_0})\geqslant \nu (z_0-B_{1-r_0})\mathbb {E}^x \tau _{D_r}. \end{aligned}$$

By Theorem 4.6, subadditivity of $V$, $\mathbb {E}^x \tau _{D_r}\geqslant \frac{ C_9}{{H_{r_0/2}}} V(r_0/2)V(\delta _{B_{1}}(x))\geqslant \frac{ C_9}{{8H_{1}}} V(r)$ $V(\delta _{B_{1}}(x))$. Since $\nu$ is isotropic, $\nu (z_0-B_{1-r_0})\geqslant c_1\nu (B_{1}\setminus B_{2r_0} ) \geqslant c_1\frac{\mathcal {I}(1)}{4V^2(r)}$, where $c_1=c_1(d)$. Therefore,

$$\begin{aligned} \mathbb {P}^x(\tau _{B_{1}}>t) \geqslant c_1\frac{ C_9}{64H_{1}}{\mathcal {I}(1)} \frac{V(\delta _{B_{1}}(x))}{V(r)}= C_{12} \frac{{\mathcal {I}(1)}}{H_1} \frac{V(\delta _{B_{1}}(x))}{\sqrt{t}}, \end{aligned}$$

where $C_{12}=c_1C_9\sqrt{C_3}/512$.

For arbitrary $R>0$ we use scaling as in the proof of Corollary 4.5. $\square$

Remark 5

The estimate in Proposition 6.1 is sharp if $t\leqslant C_{11}V^2(R)$; a reverse inequality follows immediately from Proposition 2.6. If $t>C_{11}V^2(R)$, then one can use spectral theory to observe exponential decay of the Dirichlet heat kernel and the survival probability in time if, say, $\sup _x p_t(x) <\infty$ for all $t>0$ (see [10, Corollary 7], [20, Theorem 4.2.5], [25, Theorem 3.1]).

Lemma 6.2

Let $R>0$, $D=\overline{B}^c_R$ and let $(\mathbf{H})$ hold. There is $C_{13}=C_{13}(d)$ such that,

$$\begin{aligned} \mathbb {P}^x(\tau _{D}>t)\leqslant C_{13}{\frac{H_R}{ ( \mathcal {J}(R))^{2}}}\frac{V(\delta _D(x))}{\sqrt{t}\wedge V(R)}, \qquad t>0, \; x\in {\mathbb {R}^{d}}. \end{aligned}$$

Proof

Let $x\in D$ and $x_0=xR/|x|$. If $0<t\leqslant V^2(R)$, then we choose $r$ so that $V(r)= \sqrt{t}$, otherwise we set $r=R$. We define

$$\begin{aligned} D_1= B(x_0, r)\cap B_{R}^c. \end{aligned}$$

Since $H_R\geqslant 1$ and $\mathcal {J}(R)\leqslant c(d)$, we may assume that $0<\delta _D(x)\leqslant r/2$. By Corollary 4.5,

$$\begin{aligned} \mathbb {E}^x \tau _{D_1}\leqslant C_{8} \frac{H_R}{(\mathcal {J}(R))^{2}} V(r)V(\delta _D(x)). \end{aligned}$$

By (2.25),

$$\begin{aligned} \mathbb {P}^x(|X_{\tau _{D_1}}{-x_0}|\geqslant r)\leqslant C_1 \frac{ \mathbb {E}^x \tau _{D_1}}{V^2(r)}. \end{aligned}$$

Finally, we get the conclusion:

$$\begin{aligned} \mathbb {P}^x(\tau _D>t)&\leqslant \mathbb {P}^x(\tau _{D_1}>t)+ \mathbb {P}^x(|X_{\tau _{D_1}}{-x_0}|\geqslant r) \leqslant \frac{\mathbb {E}^x \tau _{D_1}}{t}+ C_1\frac{ \mathbb {E}^x \tau _{D_1}}{V^2(r)}\\&\leqslant (C_1+1) {C_{8}H_R}(\mathcal {J}(R))^{-2}\frac{V(\delta _D(x))}{\sqrt{t}\wedge V({R})}. \end{aligned}$$

$\square$

Remark 6

If $d=1$, then regardless of $(\mathbf{H})$, we have for any $t>0$,

$$\begin{aligned} \mathbb {P}^x(\tau _{D}>t)\leqslant C_{13}\frac{V(\delta _D(x))}{\sqrt{t}\wedge V(R)}. \end{aligned}$$

This is easily seen from the above proof and the estimate $\mathbb {E}^x \tau _{D_1}\leqslant 2V(r/2)V(\delta _D(x))$. The estimate is not, however, sharp for large $t$ if $D$ is bounded.

We end this section with bounds for the survival probabilities in the complement of the ball. Noteworthy the constants in the bounds do not depend on the radius.

Theorem 6.3

Suppose that $\psi \in \mathrm WLSC ({\underline{\alpha }},0,{\underline{c}})\cap \mathrm WUSC ({\overline{\alpha }},0,{\overline{C}})$. Let $R>0$ and $D=\overline{B}^c_R$.

• (i) There is a constant $C^*=C^*(d,{\underline{\alpha }},\,{\underline{c}},\,{\overline{\alpha }},\,{\overline{C}})$ such that,

  $$\begin{aligned} \mathbb {P}^x(\tau _{D}>t)\leqslant C^*\left( \frac{V(\delta _D(x))}{\sqrt{t}\wedge V(R)}\wedge 1\right) ,\qquad t>0. \end{aligned}$$

• (ii) If $d> {\overline{\alpha }}$, then

  $$\begin{aligned} \mathbb {P}^x(\tau _{D}>t)\approx \frac{V(\delta _D(x))}{\sqrt{t}\wedge V(R)}\wedge 1, \qquad t>0, \end{aligned}$$

  where the comparability constant depends only on $d,\,{\underline{\alpha }},\,{\underline{c}},\,{\overline{\alpha }},\,{\overline{C}}$.

Proof

In the proof we make the convention that all the starred constants may only depend on $d,\,{\underline{\alpha }},\,{\underline{c}},\,{\overline{\alpha }},\,{\overline{C}}$. By Remark 6 we only need to deal with the first part only for $d\geqslant 2$. By the assumption on $\psi$ and Proposition 5.2, $\inf _{R>0}\mathcal {J}(R)\geqslant c^*_1>0$. Furthermore, for $d\geqslant 2$, by Lemma 7.2 or Lemma 7.3 we have ${H_\infty }<\infty$. The first claim now follows from Lemma 6.2.

Let $d>{\overline{\alpha }}$. By (2.21) we have absolute constant $c_2$ such that

$$\begin{aligned} \mathbb {P}^x(\tau _{D}>t)\geqslant c_2 \frac{V(\delta _D(x))}{\sqrt{t}}\wedge 1,\qquad t>0,\quad x\in {\mathbb {R}^{d}}. \end{aligned}$$

Therefore, it is enough to show that there is $c^*_3$ such that

$$\begin{aligned} \mathbb {P}^x(\tau _{D}=\infty )\geqslant c^*_3 \left( \frac{V(\delta _D(x))}{V(R)}\wedge 1\right) . \end{aligned}$$

(6.2)

Since $d>{\overline{\alpha }}$, by Corollary 5.9, there is $c^*_4\geqslant 2$ such that for $|x|\geqslant c^*_4R$, $\mathbb {P}^x(\tau _{D}=\infty )\geqslant 1/2$. It is now enough to show (6.2) for $R\leqslant |x|\leqslant 3R/2$. Let $F= B(\frac{3Rx}{2|x|},\frac{R}{2})$. By the strong Markov property,

$$\begin{aligned} \mathbb {P}^x(\tau _{D}=\infty )\geqslant \mathbb {E}^x \left( \mathbb {P}^{X_{\tau _F}} (\tau _{D}=\infty ), |X_{\tau _F}|\geqslant c^*_4R \right) \geqslant (1/2) \mathbb {P}^x ( |X_{\tau _F}|\geqslant c^*_4R ). \end{aligned}$$

By the Ikeda–Watanabe formula,

$$\begin{aligned} \mathbb {P}^x ( |X_{\tau _F}|\geqslant c^*_4R )\geqslant \nu (\{y: y_1\geqslant c^*_4 R \})\,\mathbb {E}^x\tau _F. \end{aligned}$$

By Theorem 4.1 and subadditivity of $V$ we have $\mathbb {E}^x\tau _F\geqslant c^*_5 V(\delta _D(x))V(R)$. By Lemma 5.1 and Proposition  2.4 for $X^1_t$ and subadditivity of $V$ we obtain $\nu (\{y: y_1\geqslant c^*_4 R \})\geqslant c^*_6/V^2(R)$ for some $c^*_6>0$. This proves (6.2). $\square$

We note that the assumption $d> {\overline{\alpha }}$ cannot in general be removed from the second part of the theorem. For example, if $d=1$, then the survival probability of the Cauchy process has asymptotics of logarithmic type, see [9, Remark 10]. Precise estimates of the tails of the hitting time of the ball for the isotropic stable Lévy processes are given in [9]. For the Brownian motion, [15] gives even more-precise estimates of the derivative of the survival probability.

Remark 7

We conclude this section with an obvious but necessary remark: if $B,\overline{B'}\subset {\mathbb {R}^{d}}$ are balls (open and closed, correspondingly) and $B\subset D\subset \overline{B'}^c$, then the survival probability of $D$ is bounded as follows

$$\begin{aligned} \mathbb {P}^x(\tau _{B}>t)\leqslant \mathbb {P}^x(\tau _{D}>t)\leqslant \mathbb {P}^x(\tau _{\overline{B'}^c}>t), \qquad x\in {\mathbb {R}^{d}},\, t\geqslant 0. \end{aligned}$$

This leads to immediate bounds for the survival probabilities for general $C^{1,1}$ open sets $D\subset {\mathbb {R}^{d}}$: If $\psi \in \mathrm WLSC ({\underline{\alpha }},0,{\underline{c}})\cap \mathrm WUSC ({\overline{\alpha }},0,{\overline{C}})$ and $D$ is $C^{1,1}$ at scale $r$, then by Proposition 6.1, Remark 5 and Theorem 6.3, there is $C^*=C^*(d,{\underline{\alpha }},{\overline{\alpha }},{\underline{c}},{\overline{C}})$ such that if $x\in {\mathbb {R}^{d}}$ and $t\leqslant C_{11}V^2(r)$, then

$$\begin{aligned} \frac{1}{C^*}\left( \frac{V(\delta _D(x))}{\sqrt{t}}\wedge 1\right) \leqslant \mathbb {P}^x(\tau _{D}>t)\leqslant C^*\left( \frac{V(\delta _D(x))}{\sqrt{t}}\wedge 1\right) . \end{aligned}$$

(6.3)

7 Discussion of assumptions and applications

7.1 Condition $(\mathbf{H})$

Recall that function $v>0$ is called log-concave if $\log v$ is concave, and if this is the case, then the (right hand side) derivative $v'$ of $v$ exists and $v'/v$ is non-increasing. The next lemma shows that $(\mathbf{H^*})$ is satisfied with $H_\infty =5$ if $V$ is log-concave.

Lemma 7.1

If $V$ is log-concave and ${0<}x{\leqslant y\leqslant z\leqslant }5x$, then $V({z})-V({y})\leqslant 5 V^\prime (x){(z-y)}$.

Proof

We have $V>0$ increasing, and $V'/V>0$ non-increasing. Therefore,

$$\begin{aligned} \log V(z)-\log V(y)=\int _y^z \frac{V'(s)}{V(s)}ds\leqslant \frac{V^\prime (x)}{V(x)}(z-y), \end{aligned}$$

and

$$\begin{aligned} \log V(z)-\log V(y)=\int _{V(y)}^{V(z)}\frac{1}{u}\, du\geqslant \frac{V(z)- V(y)}{V(z)}. \end{aligned}$$

By this and subadditivity of $V$,

$$\begin{aligned} V(z)- V(y)\leqslant \frac{ V(z)V^\prime (x)}{V(x)}(z-y)\leqslant 5 V^\prime (x)(z-y). \end{aligned}$$

$\square$

The next lemma shows that for dimension $d\geqslant 3$, the weak lower scaling condition implies $(\mathbf{H})$, while the weak global lower scaling implies $(\mathbf{H^*})$. This helps extend many results previously known only for complete subordinate Brownian motions with scaling (see below for definitions).

Remark 8

A sufficient condition for log-concavity of $V$ is that $V'$ be monotone, which is common for subordinate Brownian motions, for instance if the subordinator is special. For complete subordinate Brownian motions, $V$ is even a Bernstein function (see [37, Proposition 4.5]). It is interesting to note that $V'$ is not monotone for the so-called truncated $\alpha$-stable Lévy processes with $0<\alpha <2$ [32]. Indeed, if the Lévy measure has compact support, then by [21, (5.3.4)] the Lévy measure of the ladder-height process (subordinator) has compact support as well. By [44, Proposition 11.16], $\kappa$ is not a special Bernstein function, therefore by [44, Theorem 11.3], $V'$ is not decreasing. We, however, note that the truncated stable processes have global weak lower scaling with ${\underline{\alpha }}=\alpha$, and our estimates of the expected exit time for the ball hold for these processes with the comparability constant independent of $r$. This shows flexibility of our methods.

Lemma 7.2

If $d\geqslant 3$ and $\psi \in {\textit{WLSC}}(\beta , \theta , {\underline{c}})$, then $(\mathbf{H})$ holds with $H_R=H_R(\beta , \theta , {\underline{c}}, R)$ for any $R\in (0,\infty )$. If, furthermore, $\theta =0$, then $(\mathbf{H^*})$ even holds.

Proof

By [24, Corollary 5], the scale invariant Harnack inequality holds for $X^1$, the one-dimensional projection of $X$. Namely, for every $R>0$ there is $C_R<\infty$ such that if $0<r\leqslant R$, $h\geqslant 0$ on $\mathbb {R}$ and $h$ is harmonic for $X^1$ on $(-r,r)$, then

$$\begin{aligned} \sup _{y\in (-r/2,r/2)}h(y)\leqslant C_R\inf _{y\in (-r/2,r/2)}h(y). \end{aligned}$$

Since $V^\prime$ is harmonic on $(0, \infty )$ for $X^1$ and $x_0\geqslant 2r$, then by spatial homogeneity,

$$\begin{aligned} \sup _{\theta \in (x_0-r,x_0+r)}V^\prime (\theta )\leqslant C_R \inf _{\theta \in (x_0-r,x_0+r)}V^\prime (\theta ). \end{aligned}$$

Using the inequality with $(x_0,r)=(x,x/2),\,(9x/4,x),\,(4x,x)$, where $0<x\leqslant R$ we get

$$\begin{aligned} \sup _{\theta \in (x/2,5x)}V^\prime (\theta )\leqslant C^3_R \inf _{\theta \in (x/2,5x)}V^\prime (\theta )\leqslant C^3_R V^\prime (x). \end{aligned}$$

The absolute continuity of $V$ yields the conclusion. $\square$

Lemma 7.3

Let $d\geqslant 1$ and $\psi \in \mathrm WLSC ({\underline{\alpha }}, \theta , {\underline{c}})\cap \mathrm WUSC ({\overline{\alpha }}, \theta , {\overline{C}})$. Then $(\mathbf{H})$ holds with $H_R=H_R({\underline{\alpha }},{\overline{\alpha }}, \theta , {\underline{c}}, {\overline{C}},R)$ for all $R\in (0,\infty )$. If, furthermore, $\theta =0$, then $(\mathbf{H^*})$ even holds.

Proof

By the same arguments as given in Lemma 7.2 it is enough to show that the scale invariant Harnack inequality holds for $X^1$. By [10, Corollary 22 and (16)] and Proposition 2.4 applied to $X^1$, there exists $r_0>0$ such that

$$\begin{aligned} \nu _1(u)\approx \frac{1}{V(|u|)^2|u|^{d}}, \quad 0<|u|<r_0/\theta . \end{aligned}$$

At first, let $\theta >0$. By [16, Theorem 5.2] used with auxiliary function

$$\begin{aligned} \phi (r)= \left\{ \begin{array}{l@{\quad }l} V^2(r) &{} \text { if }0<r\leqslant r_0/\theta ,\\ V^2(r_0/\theta )(r\theta /r_0)^{{\overline{\alpha }}}&{} \text { if } r>r_0/\theta , \end{array} \right. \end{aligned}$$

we infer that the scale invariant Harnack inequality holds for $X^1$.

For $\theta =0$ we use [18, Theorem 4.12] instead of [16, Theorem 5.2] to get the global scale invariant Harnack inequality for $X^1$. In consequence we obtain $(\mathbf{H^*})$. $\square$

Lemma 7.4

If $\sigma >0$, then $(\mathbf{H})$ holds.

Proof

If $\sigma >0$, then $V'$ is positive, continuous and bounded by $\sigma ^{-1}$ (see Lemma 2.2). By Cauchy’s mean value theorem, for $R>0$ we have

$$\begin{aligned} V(z)-V(y)\leqslant \sigma ^{-1}(z-y)\leqslant H_RV'(x)(z-y)\quad 0<x\leqslant y\leqslant z\leqslant 5R, \end{aligned}$$

where $H_R=(\sigma \inf _{z\leqslant 5R}V'(z))^{-1}<\infty$. $\square$

The case when $X$ is a subordinate Brownian motion is of special interest in this theory: we consider a Brownian $B$ motion in ${\mathbb {R}^{d}}$ and an independent subordinator $\eta$, and we let

$$\begin{aligned} X(t)=B(2\eta (t)). \end{aligned}$$

The process $X$ is then called a subordinate Brownian motion. The monograph [44] is devoted to the study of such processes. Furthermore, $X$ is called a special subordinate Brownian motion if the subordinator is special (i.e. given by a special Bernstein function [44, Definition 11.1]), and it is called complete subordinate Brownian motion if the subordinator is even complete [44, Proposition 7.1]. We let $\varphi$ be the Laplace exponent of the subordinator, i.e.

$$\begin{aligned} \mathbb {E}\exp [-u\eta (t)]=\exp [-t\varphi (u)], \quad u\geqslant 0. \end{aligned}$$

Since

$$\begin{aligned} \mathbb {E}e^{i\langle \xi ,B_t \rangle }=e^{-t|\xi |^2/2}, \quad t\geqslant 0, \; \xi \in {\mathbb {R}^{d}}, \end{aligned}$$

we have

$$\begin{aligned} \psi (\xi )=\varphi (|\xi |^2). \end{aligned}$$

Then by [37, Theorem 4.4], $V(r)\approx \varphi (r^{-2})^{-1/2}$. For clarity, [37] makes the assumption that $\varphi$ is unbounded, but it is not necessary for the result. In connection to [37, Remark 4.7] we note that $\varphi (x)$ and $x/\varphi (x)$ are monotone. For instance, by concavity, if $s\geqslant 1$ and $x\geqslant 0$, then $\varphi (sx) \leqslant s \varphi (x)$, hence $sx/\varphi (sx) \geqslant x/\varphi (x)$.

Remark 9

If $X$ is a subordinate Brownian motion, then due to [24, Theorem 7] we may skip the assumption $d\geqslant 3$ in Lemma 7.2. This is related to the fact that Harnack inequality is inherited by orthogonal projections of isotropic unimodal Lévy processes, and every subordinate Brownian motion in dimensions $1$ and $2$ is a projection of a subordinate Brownian motion in dimension $3$ (this observation was used before in [31]).

Lemma 7.5

If $X$ is a special subordinate Brownian motion, then $V$ is concave.

Proof

By [33, Proposition 2.1], the Laplace exponent $\kappa$ given by (2.14) is a special Bernstein function. In fact, [33] makes the assumption that the Laplace exponent of a subordinator is a complete Bernstein function, but the same proof works if it is only a special Bernstein function, since it suffices that $|x|^2/\psi (x)$ be negative definite. Then [44, Theorem 11.3] implies that $V^\prime$ is non-increasing, which ends the proof. $\square$

Remark 10

Lemma 7.5 implies $(\mathbf{H^*})$ with $H_\infty =1$ for special subordinate Brownian motions.

We finish this section with a simple argument leading to boundary Harnack inequality.

Proposition 7.6

Let $\nu$ be continuous in ${\mathbb {R}^{d}{\setminus }\{0\}}$. Assume that $\psi$ satisfies the global weak lower and upper scaling conditions, $D$ is $C^{1,1}$ at scale $\rho >0$, $z\in \partial D$, $0<r<\rho$ and $u \geqslant 0$ is regular harmonic in $D\cap B(z,r)$ and vanishes in $B(z,r)\setminus D$. Then positive $c=c(d,\psi )$, $c_1=c_1(d,\psi )$ exist such that

$$\begin{aligned} \frac{u(x)}{u(y)}\leqslant c\frac{V(\delta _{D}(x))}{V(\delta _{D}(y))} \leqslant c_1\sqrt{\frac{\psi (1/\delta _{D}(y))}{\psi (1/\delta _{D}(x))}} ,\qquad x,y\in D\cap B(z,r/2). \end{aligned}$$

Proof

[10, Corollary 27] shows that the assumptions of [35] are satisfied. By [35, Lemma 5.5] we get

$$\begin{aligned}\frac{u(x)}{u(y)}\leqslant c \frac{\mathbb {E}^x\tau _{D\cap B(z,r)}}{\mathbb {E}^x\tau _{D\cap B(z,r)}},\qquad x,y\in D\cap B(z,r/2).\end{aligned}$$

for $c=c(d,\psi )$. In fact [35, Lemma 5.5] is stated only for $r<1$ but under global weak scaling conditions one can repeat arguments of [35] to obtain the global boundary Harnack inequality (see [36]). Then we estimate the expected exit time of $D\cap B(z,r)$ by using Lemma 7.3, Theorem 4.1, Corollary 4.5 and Proposition 5.2, and we obtain the first inequality. The second inequality follows from Proposition 2.4 and (3.1). $\square$

7.2 Examples

Our results apply to the following unimodal Lévy processes. In each case our sharp bounds for the expected first exit time from the ball apply and the comparability constants depend only on the dimension and the Lévy–Khintchine exponent of the process but not on the radius of the ball. Our estimates of the probability of surviving in $B_r$ and $\overline{B_r}^c$ also hold with constants independent of $r$ if the characteristic exponent of $X$ has global upper and lower scalings (see [10] for a simple discussion of scaling). If the scalings are not global, then the constants may deteriorate as $r$ increases.

Example 1

Chapter 15 of [44] lists more than one hundred cases and classes of complete Bernstein functions. All of those which are unbounded and have killing rate $0$ are covered by our results (see Lemma 7.5): we obtain sharp estimates of the expected first exit time from the ball. In fact, the comparability constants depend only on the dimension. This is, e.g., the case for Lévy process with the characteristic exponent

$$\begin{aligned} \psi (\xi )=\left[ |\xi |^{\alpha _2}+ (|\xi |^2+m)^{\alpha _3/2}-m^{\alpha _3/2}\right] ^{1-\alpha _1/2} \log ^{\alpha _1/2}(1+|\xi |^{\alpha _4}), \end{aligned}$$

where $\alpha _1,\alpha _2,\alpha _3,\alpha _4\in [0,2]$, $\alpha _1+\alpha _2+\alpha _3>0$, $\alpha _2+\alpha _3+\alpha _4>0$ and $m\geqslant 0$, and also when

$$\begin{aligned} \psi _2(\xi )=u(|\xi |)-u(0^+), \end{aligned}$$

where $u(r)=mr^2+r^2/\log ^{\alpha _1/2}(1+r^{\alpha _4})$. These include, e.g., isotropic stable process, relativistic stable process, sums of two independent isotropic stable processes (also with Gaussian component) and geometric stable processes, variance gamma processes and conjugate to geometric stable processes [44].

Example 2

Let $0<\alpha _0\leqslant \alpha _1\leqslant \cdots \leqslant 2$, $\alpha ^*=\lim _{k\rightarrow \infty }{\alpha }_k$, and define $f(r)=r^{-\alpha _{{{\lfloor }{r}{\rfloor }}}}$, $r>0$. Then $f(1/r)\in \mathrm WLSC (\alpha _0,0,1)\cap \mathrm WUSC (\alpha ^*,0,1)$, if $\alpha ^*<2$. Consider a unimodal Lévy process with Lévy density $\nu (x)=f(|x|)|x|^{-d}$, $x\ne 0$. By [10, Proposition 28], $\psi \in \mathrm WLSC (\alpha _0,0,{\underline{c}})$. For $d\geqslant 3$ by Lemma 7.2 we get $\mathbb {E}^x\tau _{B_r}\approx 1/\sqrt{f(r)f(r-|x|)}$, where $|x|<r<\infty$, and the comparability constant is independent of $r$. If $\alpha ^*<2$, then by [24, Proposition 8] $\psi \in \mathrm WUSC (\alpha ^*,0,{\overline{C}})$. The above approximation for $\mathbb {E}^x\tau _{B_r}$ is valid for $d=2$, too, cf. Lemma 7.3.

Example 3

Let $d\geqslant 3$, $\sigma \geqslant 0$, $\nu (x)=f(|x|)/|x|^d$, $x\in {\mathbb {R}^{d}{\setminus }\{0\}}$. Let $f\geqslant 0$ be non-increasing and let $\beta >0$ be such that $f(\lambda r)\leqslant c\lambda ^{-\beta }f(r)$ for $r>0$ and $\lambda >1$ (see [24, Example 2 and 48] and Lemma 7.2). So is the case for the following processes (with $\alpha ,\alpha _1\in (0,2)$): truncated stable process ($f(r)=r^{-\alpha }\mathbf 1 _{(0,1)}(r)$), tempered stable process ($f(r)=r^{-\alpha }e^{-r}$), isotropic Lamperti stable process ($f(r)=re^{\delta {r}}(e^r-1)^{-\alpha -1}$, where $\delta <\alpha +1$) and layered stable process ($f(r)=r^{-\alpha }\mathbf 1 _{(0,1)}(r)+ r^{-\alpha _1}\mathbf 1 _{[1,\infty )}(r)$).

More examples of isotropic processes with scaling may be found in [10, Section 4.1].