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Zoran Vondraček

Publications and source records attributed to Zoran Vondraček.

At least 19 recordsLinked to original sources
arXiv2026

Heat kernel estimates for Markov processes with blowing-up jump kernels

In this paper, we establish sharp two-sided heat kernel estimates for a large class of purely discontinuous symmetric Markov processes on closed subsets $F$ of $\mathbb{R}^d$, whose jump kernels blow up on a Borel subset $Σ$ of $F$. We assume that $F\setminus Σ$ is a $κ$-fat set and is dense in $F$. To the best of our knowledge, this is the first work establishing sharp heat kernel estimates for jump processes whose jump kernels blow up on part of the state space. The jump kernels under consideration take the form $J(x,y)=|x-y|^{-d-α}{\mathcal B}(x,y)$, where $α\in (0,2)$ and the function ${\mathcal B}(x,y)$ blows up at a subset $Σ$ of $F$. A fundamental obstacle is that the tails of the jump measures are not uniformly bounded, and hence standard techniques in heat kernel analysis do not provide a priori off-diagonal estimates. To overcome this difficulty, we develop a new approach based on weighted integral estimates for the heat kernel that are sensitive to both the blow-up behavior of the jump kernel and the geometry of $F\setminus Σ$. Examples of processes falling within our general framework include traces of isotropic $α$-stable processes in $C^{1,\rm Dini}$ sets, processes in Lipschitz sets arising in connection with the nonlocal Neumann problem, and a large class of resurrected self-similar processes in the closed upper half-space.

math.PR↗
arXiv2025

Heat kernel estimates for Markov processes in bounded sets with jump kernels decaying at the boundary

In this paper, we study two types of purely discontinuous symmetric Markov processes $X$ in bounded smooth subsets of $\mathbb R^d$: conservative processes and processes killed either upon approaching the boundary of the set or by a killing potential $κ$. The jump kernel of $X$ is of the form $J(x,y)={\cal B}(x,y)|x-y|^{-d-α}$, $α\in (0,2)$, where the function ${\cal B}(x,y)$ decays to 0 at the boundary and is described in terms of two $O$-regularly varying functions and one slowly varying function. Under the conditions, introduced in \cite{CKSV24}, on ${\cal B}(x,y)$ and on the killing potential $κ$, we establish sharp two-sided estimates on the heat kernel of $X$: in Lipschitz sets when $X$ is conservative, and in $C^{1,1}$ open sets for the killed process.

math.PR↗
arXiv2025

Large solutions to semilinear equations for subordinate Laplacians in $C^{1,1}$ bounded open sets

We study the existence of a large solution to a semilinear problem in a bounded open $C^{1,1}$ set for a class of nonlocal operators obtained by an appropriate subordination of the Laplacian. These operators are classical generalisations of the fractional Laplacian. The existence result is shown under a nonlocal version of the Keller-Osserman condition, stated in terms of the subordinator and the source term $f$.

math.AP↗
arXiv2025

Heat kernel estimates for Dirichlet forms degenerate at the boundary

The goal of this paper is to establish sharp two-sided estimates on the heat kernels of two types of purely discontinuous symmetric Markov processes in the upper half-space of $\mathbb R^d$ with jump kernels degenerate at the boundary. The jump kernels are of the form $J(x,y)=\mathcal B(x,y)|x-y|^{-α-d}$, $α\in (0,2)$, where the function $\mathcal B$ depends on four parameters and may vanish at the boundary. Our results are the first sharp two-sided estimates for the heat kernels of non-local operators with jump kernels degenerate at the boundary. The first type of processes are conservative Markov processes on $\overline{\mathbb R}^d_+$ with jump kernel $J(x,y)$. Depending on the regions where the parameters belong, the heat kernels estimates have three different forms, two of them are qualitatively different from all previously known heat kernel estimates. The second type of processes are the processes above killed either by a critical potential or upon hitting the boundary of the half-space. We establish that their heat kernel estimates have the approximate factorization property with survival probabilities decaying as a power of the distance to the boundary, where the power depends on the constant in the critical potential.

math.PR↗
arXiv2025

On the nonlocal heat equation for certain Lévy operators and the uniqueness of positive solutions

We develop a Widder-type theory for nonlocal heat equations involving quite general Lévy operators. Thus, we consider nonnegative solutions and look for conditions on the operator that ensure: (i) uniqueness of nonnegative classical and very weak solutions with a given initial trace; (ii) the existence of an initial trace, belonging to certain admissibility class; and (iii) the existence of a solution, given by a representation formula, for any admissible initial trace. Such results are obtained first for purely nonlocal Lévy operators defined through positive symmetric Lévy kernels comparable to radial functions with mixed polynomial growth, and then extended to more general operators, including anisotropic ones and operators that have both a local and a nonlocal part.

math.AP↗
arXiv2025

Potential theory of Dirichlet forms with jump kernels blowing up at the boundary

In this paper we study the potential theory of Dirichlet forms on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-α}\mathcal{B}(x,y)$ and the killing potential $κx_d^{-α}$, where $α\in (0, 2)$ and $\mathcal{B}(x,y)$ can blow up to infinity at the boundary. The jump kernel and the killing potential depend on several parameters. For all admissible values of the parameters involved and all $d \ge 1$, we prove that the boundary Harnack principle holds, and establish sharp two-sided estimates on the Green functions of these processes.

math.PR↗
arXiv2024

Markov processes with jump kernels decaying at the boundary

The goal of this work is to develop a general theory for non-local singular operators of the type $$ L^{\mathcal{B}}_αf(x)=\lim_{ε\to 0} \int_{D,\, |y-x|>ε}\big(f(y)-f(x)\big) \mathcal{B}(x,y)|x-y|^{-d-α}\,dy, $$ and $$ L f(x)=L^{\mathcal{B}}_αf(x) - κ(x) f(x), $$ in case $D$ is a $C^{1,1}$ open set in $\mathbb{R}^d$, $d\ge 2$. The function $\mathcal{B}(x,y)$ above may vanish at the boundary of $D$, and the killing potential $κ$ may be subcritical or critical. From a probabilistic point of view we study the reflected process on the closure $\overline{D}$ with infinitesimal generator $L^{\mathcal{B}}_α$, and its part process on $D$ obtained by either killing at the boundary $\partial D$, or by killing via the killing potential $κ(x)$. The general theory developed in this work (i) contains subordinate killed stable processes in $C^{1,1}$ open sets as a special case, (ii) covers the case when $\mathcal{B}(x,y)$ is bounded between two positive constants and is well approximated by certain Hölder continuous functions, and (iii) extends the main results known for the half-space in $\mathbb{R}^d$. The main results of the work are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates. Our results on the boundary Harnack principle completely cover the corresponding earlier results in the case of half-space. Our Green function estimates extend the corresponding earlier estimates in the case of half-space to bounded $C^{1, 1}$ open sets.

math.PR↗
arXiv2023

Positive self-similar Markov processes obtained by resurrection

In this paper we study positive self-similar Markov processes obtained by (partially) resurrecting a strictly $α$-stable process at its first exit time from $(0,\infty)$. We construct those processes by using the Lamperti transform. We explain their long term behavior and give conditions for absorption at 0 in finite time. In case the process is absorbed at 0 in finite time, we give a necessary and sufficient condition for the existence of a recurrent extension. The motivation to study resurrected processes comes from the fact that their jump kernels may explode at zero. We establish sharp two-sided jump kernel estimates for a large class of resurrected stable processes.

math.PR↗
arXiv2023

Harnack inequality and interior regularity for Markov processes with degenerate jump kernels

In this paper we study interior potential-theoretic properties of purely discontinuous Markov processes in proper open subsets $D\subset \mathbb{R}^d$. The jump kernels of the processes may be degenerate at the boundary in the sense that they may vanish or blow up at the boundary. Under certain natural conditions on the jump kernel, we show that the scale invariant Harnack inequality holds for any proper open subset $D\subset \mathbb{R}^d$ and prove some interior regularity of harmonic functions. We also prove a Dynkin-type formula and several other interior results.

math.PR↗
arXiv2022

On potential theory of Markov processes with jump kernels decaying at the boundary

Motivated by some recent potential theoretic results on subordinate killed Lévy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is the Lévy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal Lévy process) in $\mathbb{R}^d$. The main novelty is that the term $\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\mathcal{B}(x,y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on four parameters, $β_1, β_2, β_3$, $β_4$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialise to the case of the half-space $D=\mathbb{R}_+^d=\{x=(\widetilde{x},x_d):\, x_d>0\}$, the $α$-stable kernel $j(|x-y|)=|x-y|^{-d-α}$ and the killing function$κ(x)=c x_d^{-α}$, $α\in (0,2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(α-1)_+$, there are values of the parameters $β_1, β_2, β_3$, $β_4$, and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $β_1, β_2, β_3$, $β_4$, for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid.

math.PR↗
arXiv2022

Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary

In this paper we continue our investigation of the potential theory of Markov processes with jump kernels decaying at the boundary. To be more precise, we consider processes in ${\mathbb R}^d_+$ with jump kernels of the form ${\mathcal B}(x,y) |x-y|^{-d-α}$ and killing potentials $κ(x)=cx_d^{-α}$, $0<α<2$. The boundary part ${\mathcal B}(x,y)$ is comparable to the product of three terms with parameters $β_1, β_2$, $β_3$ and $β_4$ appearing as exponents in these terms. The constant $c$ in the killing term can be written as a function of $α$, ${\mathcal B}$ and a parameter $p\in ((α-1)_+, α+β_1)$, which is strictly increasing in $p,$ decreasing to $0$ as $p\downarrow (α-1)_+$ and increasing to $\infty$ as $p\uparrowα+β_1$. We establish sharp two-sided estimates on the Green functions of these processes for all $p\in ((α-1)_+, α+β_1)$ and all admissible values of $β_1, β_2$, $β_3$ and $β_4$. Depending on the regions where $β_1$, $β_2$ and $p$ belong, the estimates on the Green functions are different. In fact, the estimates have three different forms depending on the regions the parameters belong to. As applications, we prove that the boundary Harnack principle holds in certain region of the parameters and fails in some other region of the parameters. Combined with the main results of \cite{KSV},we completely determine the region of the parameters where the boundary Harnack principle holds.

math.PR↗
arXiv2022

Potential theory of Dirichlet forms degenerate at the boundary: the case of no killing potential

In this paper we consider the Dirichlet form on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-α}\mathcal{B}(x,y)$, where $\mathcal{B}(x,y)$ can be degenerate at the boundary. Unlike our previous works [6,7] where we imposed critical killing, here we assume that the killing potential is identically zero. In case $α\in (1,2)$ we first show that the corresponding Hunt process has finite lifetime and dies at the boundary. Then, as our main contribution, we prove the boundary Harnack principle and establish sharp two-sided Green function estimates. Our results cover the case of the censored $α$-stable process, $α\in (1,2)$, in the half-space studied in [2].

math.PR↗
arXiv2022

Heat kernel estimates for subordinate Markov processes and their applications

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates.

math.PR↗
arXiv2020

Factorization and estimates of Dirichlet heat kernels for non-local operators with critical killings

In this paper we discuss non-local operators with killing potentials, which may not be in the standard Kato class. We first discuss factorization of their Dirichlet heat kernels in metric measure spaces. Then we establish explicit estimates of the Dirichlet heat kernels under critical killings in $C^{1,1}$ open subsets of $\mathbb{R}^d$ or in $\mathbb{R}^d\setminus\{0\}$. The decay rates of our explicit estimates come from the values of the multiplicative constants in the killing potentials. Our method also provides an alternative and unified proof of the main results of \cite{CKS10a, CKS10b, CKS-AOP}.

math.PR↗
arXiv2019

A probabilistic approach to a non-local quadratic form and its connection to the Neumann boundary condition problem

In this paper, we look at a probabilistic approach to a non-local quadratic form that has lately attracted some interest. This form is related to a recently introduced non-local normal derivative. The goal is to construct two Markov process: one corresponding to that form and the other which is related to a probabilistic interpretation of the Neuman problem. We also study the Dirichlet-to-Neumann operator for non-local operators.

math.PR↗
arXiv2019

Potential theory of subordinate killed Brownian motion

Let $W^D$ be a killed Brownian motion in a domain $D\subset {\mathbb R}^d$ and $S$ an independent subordinator with Laplace exponent $ϕ$. The process $Y^D$ defined by $Y^D_t=W^D_{S_t}$ is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator $-ϕ(-Δ|_D)$, where $Δ|_D$ is the Dirichlet Laplacian. In this paper we study the potential theory of $Y^D$ under a weak scaling condition on the derivative of $ϕ$. We first show that non-negative harmonic functions of $Y^D$ satisfy the scale invariant Harnack inequality. Subsequently we prove two types of scale invariant boundary Harnack principles with explicit decay rates for non-negative harmonic functions of $Y^D$. The first boundary Harnack principle deals with a $C^{1,1}$ domain $D$ and non-negative functions which are harmonic near the boundary of $D$, while the second one is for a more general domain $D$ and non-negative functions which are harmonic near the boundary of an interior open subset of $D$. The obtained decay rates are not the same, reflecting different boundary and interior behaviors of $Y^D$.

math.PR↗
arXiv2019

On the boundary theory of subordinate killed Lévy processes

Let $Z$ be a subordinate Brownian motion in ${\mathbb R}^d$, $d\ge 2$, via a subordinator with Laplace exponent $ϕ$. We kill the process $Z$ upon exiting a bounded open set $D\subset {\mathbb R}^d$ to obtain the killed process $Z^D$, and then we subordinate the process $Z^D$ by a subordinator with Laplace exponent $ψ$. The resulting process is denoted by $Y^D$. Both $ϕ$ and $ψ$ are assumed to satisfy certain weak scaling conditions at infinity. We study the potential theory of $Y^D$, in particular the boundary theory. First, in case that $D$ is a $κ$-fat bounded open set, we show that the Harnack inequality holds. If, in addition, $D$ satisfies the local exterior volume condition, then we prove the Carleson estimate. In case $D$ is a smooth open set and the lower weak scaling index of $ψ$ is strictly larger than $1/2$, we establish the boundary Harnack principle with explicit decay rate near the boundary of $D$. On the other hand, when $ψ(λ)=λ^γ$ with $γ\in (0,1/2]$, we show that the boundary Harnack principle near the boundary of $D$ fails for any bounded $C^{1,1}$ open set $D$. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not. One of the main ingredients in the proofs is the sharp two-sided estimates of the Green function of $Y^D$. Under an additional condition on $ψ$, we establish sharp two-sided estimates of the jumping kernel of $Y^D$ which exhibit some unexpected boundary behavior. We also prove a boundary Harnack principle for non-negative functions harmonic in a smooth open set $E$ strictly contained in $D$, showing that the behavior of $Y^D$ in the interior of $D$ is determined by the composition $ψ\circ ϕ$.

math.PR↗