Heat kernel estimates for the fractional Laplacian with Dirichlet conditions

Abstract

We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.

Global Heat Kernel Estimates for Relativistic Stable Processes in Half-space-like Open Sets

“…For the upper bound estimate in a half-space, we use some results in [21]. Then, to get the full two-sided heat kernel estimates, we adapt the strategy in [2,7] to deal with large time estimates.…”

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

“…A more recent strand concerns jump processes living in bounded domains in Euclidean space with boundaries of different degrees of regularity. Exemplary investigations using sophisticated probabilistic techniques cover the special cases of fractional Laplacians [8,10] and Vol. 87 (2017) Off-diagonal heat kernel asymptotics... 329 the relativistic stable processes [11].…”

Heat kernel estimates for Δ+Δ α /2 in C 1, 1 open sets

“…The proof presented here is more direct, and uses only the continuity of λ a, D 1 and its corresponding first eigenfunction in a ∈ (0, M], which is established in [18]. Lastly, we point out that the approach of [3] relies critically on the fact that symmetric stable processes do not have a diffusion component and so it is not directly applicable to the processes considered in this paper.…”

“…In [19], the large time behaviors of heat kernels for symmetric α-stable processes and censored stable processes in unbounded open sets were studied. Very recently, in [2,3], the heat kernel of the fractional Laplacian in a non-smooth open set was discussed. We refer the readers to [8] for a survey of recent progress in the heat kernel estimates of jump Markov processes.…”