相关论文: Upper Rate Functions of Brownian Motion Type for S…

In this paper, we consider a symmetric pure jump Markov process $X$ on a metric measure space with volume doubling conditions. Our focus is on estimating the transition density $p(t,x,y)$ of $X$ and studying its stability when the jumping…

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…

We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the H\"older continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.

We show two-sided bounds of heat kernel for anisotropic non-singular symmetric pure jump Markov process whose jump kernel $J(x,y)$ is comparable to $\frac{{\bf 1}_{\mathcal{V}}(x-y)}{|x-y|^{d+\alpha}}$, where $\mathcal{V}$ is a union of…

In this paper, we study the transition densities of pure-jump symmetric Markov processes in $ {{\mathbb R}}^d$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions…

We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds…

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the…

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…

We consider the symmetric non-local Dirichlet form $(E, F)$ given by \[ E (f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $F$ the closure of the set of $C^1$ functions on $R^d$ with compact support with respect to $E_1$,…

In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their…

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…

In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior $C^{1,\eta}$ open sets for all $t> 0$. The processes are symmetric pure jump Markov processes with jumping kernel…

Let $Z=(Z^{1}, \ldots, Z^{d})$ be the d-dimensional L\'evy {process} where {$Z^i$'s} are independent 1-dimensional L\'evy {processes} with identical jumping kernel $ \nu^1(r) =r^{-1}\phi(r)^{-1}$. Here $\phi$ is {an} increasing function…

We determine the decay rate of the bottom crossing probability for symmetric jump processes under the condition on heat kernel estimates. Our results are applicable to symmetric stable-like processes and stable-subordinated diffusion…

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…

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…

Let $Z=(Z^{1}, \ldots, Z^{d})$ be the $d$-dimensional L\'evy processes where $Z^{i}$'s are independent $1$-dimensional L\'evy processes with jump kernel $J^{\phi, 1}(u,w) =|u-w|^{-1}\phi(|u-w|)^{-1}$ for $u, w\in \mathbb R$. Here $\phi$ is…

By making full use of heat kernel estimates, we establish the integral tests on the zero-one laws of upper and lower bounds for the sample path ranges of symmetric Markov processes. In particular, these results concerning on upper rate…

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in $C^{1,\eta}$ open sets. The processes are symmetric pure jump Markov processes with jumping intensity $\kappa(x,y) \psi_1…

We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in (0, 1]\}$ on $\bR^d$ for every $d\geq 1$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $\alpha \in (0, 2)$.…