In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the...

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...

This paper presents an experimental and numerical study that deals with the problem of mixing grains falling down through a bi-dimensional Galton board (BGB). The special issue addressed here is the influence of the presence of lateral walls in the BGB. Disks of equal diameters but different...

For α∈(0,2) and M0, we consider a family of nonlocal operators {Δ+aαΔα/2,a∈(0,M]} on Rd under Kato class gradient perturbation. We establish the existence and uniqueness of their fundamental solutions, and derive their sharp two-sided estimates. The estimates give explicit dependence on...

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in C1,η open sets. The processes are symmetric pure jump Markov processes with jumping intensity κ(x,y)ψ1(|x−y|)−1|x−y|−d−α, where α∈(0,2). Here, ψ1 is an increasing...

We prove the existence of the integrated density of states for subordinate Brownian motions in the presence of the Poissonian random potentials on the Sierpiński gasket.

Consider a centred random walk in dimension one with a positive finite variance σ2, and let τB be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic Px(τBn)∼2/πσ−1VB(x)n−1/2 and provide an explicit formula for the limit VB as a function 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 (2002) [3] and Bogdan–Sztonyk (2005) [6] to more general processes. Furthermore, we establish a new version of the Harnack...

We consider a positive recurrent Markov chain on R+ with asymptotically zero drift which behaves like −c1/x at infinity; this model was first considered by Lamperti. We are interested in tail asymptotics for the stationary measure. Our analysis is based on construction of a harmonic function...