arXiv subjects

Zoran Vondraček

Publications and source records attributed to Zoran Vondraček.

29 records · Page 2Linked to original sources

arXiv2017

On purely discontinuous additive functionals of subordinate Brownian motions

Let $A_t=\sum_{s\le t} F(X_{s-},X_s)$ be a purely discontinuous additive functional of a subordinate Brownian motion $X=(X_t, \mathbb P_x)$. We give a sufficient condition on the non-negative function $F$ that guarantees that finiteness of $A_{\infty}$ implies finiteness of its expectation. This result is then applied to study the relative entropy of $\mathbb P_x$ and the probability measure induced by a purely discontinuous Girsanov transform of the process $X$. We prove these results under the weak global scaling condition on the Laplace exponent of the underlying subordinator.

math.PR↗

arXiv2017

Heat kernels of non-symmetric jump processes: beyond the stable case

Let $J$ be the Lévy density of a symmetric Lévy process in $\mathbb{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator $$ {\mathcal L}^κf(x):= \lim_{ε\downarrow 0} \int_{\{z \in \mathbb{R}^d: |z|>ε\}}(f(x+z)-f(x))κ(x,z)J(z)\, dz\, , $$ where $κ(x,z)$ is a Borel measurable function on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<κ_0\le κ(x,z)\le κ_1$, $κ(x,z)=κ(x,-z)$ and $|κ(x,z)-κ(y,z)|\le κ_2|x-y|^β$ for some $β\in (0, 1)$. We construct the heat kernel $p^κ(t, x, y)$ of ${\mathcal L}^κ$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^κ$.

math.PR↗

arXiv2015

Absolute continuity and singularity of probability measures induced by a purely discontinuous Girsanov transform of a stable process

In this paper we study mutual absolute continuity and singularity of probability measures on the path space which are induced by an isotropic stable Lévy process and the purely discontinuous Girsanov transform of this process. We also look at the problem of finiteness of the relative entropy of these measures. An important tool in the paper is the question under which circumstances the a.s. finiteness of an additive functional at infinity implies the finiteness of its expected value.

math.PR↗

arXiv2014

A distributional equality for suprema of spectrally positive Lévy processes

Let $Y$ be a spectrally positive Lévy process with $E Y_1<0$, $C$ an independent subordinator with finite expectation, and $X=Y+C$. A curious distributional equality proved in Huzak et al., Ann. Appl. Probab. 14 (2004) 1278--1397, states that if $E X_1<0$, then $\sup_{0\le t <\infty}Y_t$ and the supremum of $X$ just before the first time its new supremum is reached by a jump of $C$ have the same distribution. In this paper we give an alternative proof of an extension of this result and offer an explanation why it is true.

math.PR↗

arXiv2014

Minimal thinness with respect to symmetric Lévy processes

Minimal thinness is a notion that describes the smallness of a set at a boundary point. In this paper, we provide tests for minimal thinness at finite and infinite minimal Martin boundary points for a large class of purely discontinuous symmetric Lévy processes.

math.PR↗

arXiv2014

Martin boundary for some symmetric Lévy processes

In this paper we study the Martin boundary of open sets with respect to a large class of purely discontinuous symmetric Lévy processes in ${\mathbb R}^d$. We show that, if $D\subset {\mathbb R}^d$ is an open set which is $κ$-fat at a boundary point $Q\in \partial D$, then there is exactly one Martin boundary point associated with $Q$ and this Martin boundary point is minimal.

math.PR↗

arXiv2013

Unavodiable collections of balls for isotropic Lévy processes

A collection $\{\bar{B}(x_n,r_n)\}_{n\ge 1}$ of pairwise disjoint balls in the Euclidean space $\R^d$ is said to be avoidable with respect to a transient process $X$ if the process with positive probability escapes to infinity without hitting any ball. In this paper we study sufficient and necessary conditions for avoidability with respect to unimodal isotropic Lévy processes satisfying a certain scaling hypothesis. These conditions are expressed in terms of the characteristic exponent of the process, or alternatively, in terms of the corresponding Green function. We also discuss the same problem for a random collection of balls. The results are generalization of several recent results for the case of Brownian motion.

math.PR↗

arXiv2013

Unavoidable collections of balls for censored stable processes

We study avoidability of collections of balls in bounded $C^{1,1}$ opens sets for censored $α$-stable processes, $α\in (1,2)$. The results are analog to the ones obtained for Brownian motion in S. J. Gardiner, M. Ghergu, Champagne subregions of the unit ball with unavoidable bubbles, Ann. Acad. Sci. Fenn. Math. 35 (2010) 321-329. On the way we derive a Wiener-Aikawa-type criterion for minimal thinness with respect to the censored stable processes.

math.PR↗

arXiv2013

Global uniform boundary Harnack principle with explicit decay rate and its application

In this paper, we consider a large class of subordinate Brownian motions $X$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero and at infinity. We first discuss how such conditions govern the behavior of the subordinator and the corresponding subordinate Brownian motion for both large and small time and space. Then we establish a global uniform boundary Harnack principle in (unbounded) open sets for the subordinate Brownian motion. When the open set satisfies the interior and exterior ball conditions with radius $R >0$, we get a global uniform boundary Harnack principle with explicit decay rate. Our boundary Harnack principle is global in the sense that it holds for all $R>0$ and the comparison constants do not depend on $R$, and it is uniform in the sense that it holds for all balls with radii $r \le R$ and the comparison constants depend neither on $D$ nor on $r$. As an application, we give sharp two-sided estimates for the transition density and the Green function of such subordinate Brownian motions in the half-space.

math.PR↗

arXiv2012

Boundary Harnack principle and Martin boundary at infinity for subordinate Brownian motions

In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of $κ$-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.

math.PR↗

arXiv2009

Boundary Harnack principle for $Δ+ Δ^{α/2}$

For $d\geq 1$ and $α\in (0, 2)$, consider the family of pseudo differential operators $\{Δ+ b Δ^{α/2}; b\in [0, 1]\}$ on $\R^d$ that evolves continuously from $Δ$ to $Δ+ Δ^{α/2}$. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to $Δ+b Δ^{α/2}$ (or equivalently, the sum of a Brownian motion and an independent symmetric $α$-stable process with constant multiple $b^{1/α}$) in $C^{1, 1}$ open sets. Here a "uniform" BHP means that the comparing constant in the BHP is independent of $b\in [0, 1]$. Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to $Δ+ b Δ^{α/2}$ in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.

math.PR↗