Asymptotic Behaviour and Estimates of Slowly Varying Convolution Semigroups

Abstract

The authors dedicate this work to the memory of Ante Mimica.Abstract. We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on R d under the assumption that its Lévy-Khintchine exponent varies slowly. We also derive some new estimates of the transition densities and Green functions.2010 Mathematics Subject Classification. Primary 47D06, 60J75; Secondary: 44A10, 46F12.

Gradient estimates of Dirichlet heat kernels for unimodal Lévy processes

“…The following estimate follows from [, Proposition 5.3]. Lemma For any we have where , .…”

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

“…• Ψ belongs to de Hahn's class at 0 (and thus β = 0; see [2,18] for definitions), then ν has lower index −d − β at infinity.…”

“…2011/03/D/ST1/00311. and 3.5 in [21]; for the potential kernel, we refer to Theorem 3 in [16], Theorem 5.8 in [18], Proposition 4.5 in [22], and Theorem 3.2 in [25]. Theorem 1.1.…”

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

“…To the authors' best knowledge, our result (Theorems 1.5 and 1.6) is the first result on the Dirichlet heat kernel estimates for Lévy processes with low intensity of small jumps both in small time and large time. Our paper is motivated by the recent paper [25] where sharp two-sided estimates on the heat kernel in the whole space R d is established for pure jump isotropic unimodal Lévy processes (without killing) having the Lévy measure in the form ν(dx) = |x| −d (|x| −1 )dx for a bounded function slowly varying at infinity. Unlike [25], in this paper, we allow the function to be unbounded and not slowly varying at infinity.…”

“…Our paper is motivated by the recent paper [25] where sharp two-sided estimates on the heat kernel in the whole space R d is established for pure jump isotropic unimodal Lévy processes (without killing) having the Lévy measure in the form ν(dx) = |x| −d (|x| −1 )dx for a bounded function slowly varying at infinity. Unlike [25], in this paper, we allow the function to be unbounded and not slowly varying at infinity. Hence, our result is even new for the whole space case.…”