Estimates of Dirichlet heat kernels for unimodal Lévy processes with low intensity of small jumps

Abstract

In this paper, we study transition density functions for pure jump unimodal Lévy processes killed upon leaving an open set D. Under some mild assumptions on the Lévy density, we establish two-sided Dirichlet heat kernel estimates when the open set D is C 1,1 . Our result covers the case that the Lévy densities of unimodal Lévy processes are regularly varying functions whose indices are equal to the Euclidean dimension. This is the first results on two-sided Dirichlet heat kernel estimates for Lévy processes such that the lower scaling index of the Lévy densities is not necessarily strictly bigger than the Euclidean dimension.

An $L_q(L_p)$-theory for time-fractional diffusion equations with nonlocal operators generated by Lévy processes with low intensity of small jumps

“…(iv) Heat kernel bounds for small time can be shown without the condition (2.7) (see [18,12]). To obtain upper bound of fundamental solution, our approach also needs an upper bound of heat kernel for large time.…”

“…Proof. If ℓ satisfies Assumption 2.12 (i), then (3.4) is a direct consequence of [12,Corollary 2.13]. If we define…”