# 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.

## Keywords

Lévy–Khintchine exponent Unimodal isotropic Lévy process Lévy measure First exit time Survival probability Superharmonic function## Mathematics Subject Classification (2010)

Primary 31B25 60J50 Secondary 60J75 60J35## 1 Introduction

*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

*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*.

*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),

*sharp*, meaning that the ratio of its extreme sides is bounded. One of our main contributions is the following inequality,

*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].

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.

*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 \(D\):

*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*,

*diameter*,

*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 \({\mathbb {R}^{d}}\).

*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

*regular*harmonic in \(D\) if (2.2) holds for \(U=D\).

### 2.1 Isotropic Lévy processes

*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

### **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*

### *Remark 1*

*resolvent measures*

### 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 \(\psi \).

*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 \(\sigma >0\).

*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**

### *Proof*

### **Lemma 2.5**

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

### *Proof*

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**

### *Proof*

### *Remark 2*

### **Lemma 2.7**

### *Proof*

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

### **Corollary 2.8**

### *Proof*

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*

### *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].

*killed off*open \(D{\subset {\mathbb {R}^{d}}}\) is defined by Hunt’s formula,

## 3 Barriers for unimodal Lévy processes

### **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*

### **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*

*all*\(r>0\). Let \(r_1>0\) be such that \(D\subset B_{r_1/2}\), and let \(r\geqslant r_1\). We have

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**

### *Proof*

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**

### *Proof*

### **Lemma 3.7**

### *Proof*

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

### **Definition 1**

- 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).

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**

### *Proof*

### **Lemma 3.9**

### *Proof*

## 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**

### *Proof*

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**

### *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**

### *Proof*

### **Lemma 4.4**

### *Proof*

### **Corollary 4.5**

### *Proof*

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**

### *Proof*

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

*weak lower scaling*condition (at infinity) if there are numbers \({\underline{\alpha }}>0\) and \({\underline{c}}{\in (0,1]}\), such that

*global*weak lower scaling condition.

*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**

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*

*complete Bernstein function*:

### **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*

### 5.1 Hitting a ball

### **Lemma 5.4**

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

### **Lemma 5.5**

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**

### *Proof*

### **Lemma 5.7**

### *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**

### *Proof*

The following result is important in Sect. 6.

### **Corollary 5.9**

## 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**

### *Proof*

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**

### *Proof*

### *Remark 6*

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**

- (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 \(d> {\overline{\alpha }}\), thenwhere the comparability constant depends only on \(d,\,{\underline{\alpha }},\,{\underline{c}},\,{\overline{\alpha }},\,{\overline{C}}\).$$\begin{aligned} \mathbb {P}^x(\tau _{D}>t)\approx \frac{V(\delta _D(x))}{\sqrt{t}\wedge V(R)}\wedge 1, \qquad t>0, \end{aligned}$$

### *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.

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*

## 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*

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*

### **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*

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*

*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.

### *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**

### *Proof*

### 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*

### *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].

## Notes

### Acknowledgments

Tomasz Grzywny was supported by the Alexander von Humboldt Foundation and expresses his gratitude for the hospitality of Technische Unversität Dresden and Unversität Bielefeld, where the paper was written in main part. We thank the referees for very useful comments and suggestions.

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