Related papers: Boundary Harnack inequality for Markov processes w…

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…

We prove existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular domains, in the context of general metric measure spaces. As a corollary, we prove uniqueness of…

Motivated by some recent potential theoretic results on subordinate killed L\'evy 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…

We prove regularity estimates for functions which are harmonic with respect to certain jump processes. The aim of this article is to extend the method of Bass-Levin[BL02] and Bogdan-Sztonyk[BS05] to more general processes. Furthermore, we…

In this paper, we study purely discontinuous symmetric Markov processes on closed subsets of ${\mathbb R}^d$, $d\ge 1$, with jump kernels of the form $J(x,y)=|x-y|^{-d-\alpha}{\mathcal B}(x,y)$, $\alpha\in (0,2)$, where the function…

In a setting, where only "exit measures" are given, as they are associated with an arbitrary right continuous strong Markov process on a separable metric space, we provide simple criteria for the validity of Harnack inequalities for…

We establish a boundary Harnack principle for a large class of subordinate Brownian motion, including mixtures of symmetric stable processes, in bounded $\kappa$-fat open set (disconnected analogue of John domains). As an application of the…

In this paper we prove the uniform boundary Harnack principle in general open sets for harmonic functions with respect to a large class of rotationally symmetric purely discontinuous L\'evy processes.

We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jumping measure in the recent characterization of the parabolic…

Let $X$ be an isotropic unimodal L\'{e}vy jump process on $\mathbb{R}^d$. We develop probabilistic methods which in many cases allow us to determine whether $X$ satisfies the elliptic Harnack inequality (EHI), by looking only at the jump…

We derive fluctuation-response inequalities for Markov jump processes that link the fluctuations of general observables to the response to perturbations in the transition rates within a unified framework. These inequalities are derived…

Given a symmetric diffusion process and a jump process on the same underlying space, is there a subordinator such that the jump process and the subordinated diffusion processes are comparable? We address this question when the diffusion…

The goal of this work is to develop a general theory for non-local singular operators of the type $$ L^{\mathcal{B}}_{\alpha}f(x)=\lim_{\epsilon\to 0} \int_{D,\, |y-x|>\epsilon}\big(f(y)-f(x)\big) \mathcal{B}(x,y)|x-y|^{-d-\alpha}\,dy, $$…

In this paper, a necessary and sufficient condition is obtained for the scale invariant boundary Harnack inequality (BHP in abbreviation) for a large class of Hunt processes on metric measure spaces that are in weak duality with another…

Consider a symmetric Markovian jump process $\{X_t\}$ on a metric measure space $(M, d, \mu)$. Chen, Kumagai, and Wang recently showed that two-sided heat kernel estimates and the parabolic Harnack inequality are both stable under bounded…

We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal L\'evy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in $C^{1,1}$ open…

We consider a model of Brownian motion on a bounded open interval with instantaneous jumps. The jumps occur at a spatially dependent rate given by a positive parameter times a continuous function positive on the interval and vanishing on…

In this paper, we consider a product of a symmetric stable process in $\mathbb{R}^d$ and a one-dimensional Brownian motion in $\mathbb{R}^+$. Then we define a class of harmonic functions with respect to this product process. We show that…

We characterize Gaussian estimates for transition probability of a discrete time Markov chain in terms of geometric properties of the underlying state space. In particular, we show that the following are equivalent: (1) Two sided Gaussian…

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…