Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in C^1,η open sets

2014/02/19 by Kyung-Youn Kim, Kim, Kyung-Youn, Panki Kim +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Mathematical Approximation and Integration #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.1402.4660

Abstract

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in C^1,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 function on [ 0, ∞ ), with ψ₁(r)=1 on 01 for β∈ [0,∞], and κ( x, y) is a symmetric function confined between two positive constants, with |κ(x,y)-κ(x,x)|≤ c₅|x-y|^ρ for |x-y|<1 and ρ>α/2. We establish two-sided estimates for the transition densities of such processes in C^1,η open sets when η∈ (α/2, 1]. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in C^1,η open sets when η∈ (α/2, 1].

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