Dirichlet heat kernel estimates for rotationally symmetric Lévy processes

By Zhen-Qing Chen, Panki Kim, Renming Song. 2013

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In this paper, we consider a large class of purely discontinuous rotationally symmetric Lévy processes. We establish sharp two‐sided estimates for the transition densities of such processes killed upon leaving an open set D . When D is a κ ‐fat open set, the sharp two‐sided estimates are given in terms of surviving probabilities and the global transition density of the Lévy process. When D is a C1,1 open set and the Lévy exponent of the process is given by Ψ(ξ)=ϕ(|ξ|2) with ϕ being a complete Bernstein function satisfying a mild growth condition at infinity, our two‐sided estimates are explicit in terms of Ψ , the distance function to the boundary of D and the Lévy density of X . This gives an affirmative answer to the conjecture posted in Chen, Kim and Song [Global heat kernel estimates for relativistic stable processes in half‐space‐like open sets. Potential Anal. 36 (2012) 235–261]. Our results are the first sharp two‐sided Dirichlet heat kernel estimates for a large class of symmetric Lévy processes with general Lévy exponents. We also derive an explicit lower bound estimate for symmetric Lévy processes on Rd in terms of their Lévy exponents.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b81a965246e85bb56e0f2edb457ef9c74a7b0745

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Gradient estimates of harmonic functions and transition densities for Levy processes (2013, background): “ δD(x) ∧ |x− y|∧1 , x∈ D, (3) where δD(x) = dist(x,∂D). The potential theory of L´evy processes, especially subordinate Brownian motions, has attracted a lot of attention during recent years see e.g. [6, 10, 9, 18, 19]. Our study is in the scope of this type of research. The assertion of Theorem 1.1”