Heat Kernel Upper Estimates for Symmetric Jump Processes with Small Jumps of High Intensity

Abstract

We consider the following non-local operatorand β ∈ (0, 1]. We prove upper estimates for the transition density of the associated symmetric Markov jump process X. Examples of Lévy processes with generator of the type above are studied.

Estimates of densities for Lévy processes with lower intensity of large jumps

“…We do not assume here anything (except of (1)) on the behavior of ν outside of the ball B(0, 1) and we consider the processes with high intensity of small jumps, i.e., such that ν(dx) ≍ |x| −d−2 log 2 |x| −β dx, for |x| < 1, where β > 1. We extend the results obtained previously in [23] and [17]. In this case sharp estimates for large times were already known.…”

Estimates of transition densities and their derivatives for jump Lévy processes

“…for |x| < 1, t < 1, which can be shown similarly as (36) in the proof of Theorem 4, and so (33) improves the results of [20]. Exact estimate in this case is still an open question.…”

“…We also include here processes with intensities of small jumps remarkably different than the stable one. The time-space asymptotics of the densities for this class of processes is still very little understood (see [20,21]). The other novelty here are the estimates of the derivatives of the densities.…”

Small-time sharp bounds for kernels of convolution semigroups

“…Recent papers [42,43,28,30,26,27] contain the estimates for more general classes of Lévy processes, including tempered processes with intensities of jumps lighter than polynomial. The paper [6] focuses on the estimates of densities for isotropic unimodal Lévy processes with Lévy-Khintchine exponents having the weak local scaling at infinity, while the papers [33,26] discuss the processes with higher intensity of small jumps, remarkably different than the stable one. In [11,12,25] the authors investigate the case of more general, non-necessarily space homogeneous, symmetric jump Markov processes with jump intensities dominated by those of isotropic stable processes.…”