Transition Density Estimates for a Class of Lévy and Lévy-Type Processes

Abstract

We show on-and off-diagonal upper estimates for the transition densities of symmetric Lévy and Lévy-type processes. To get the on-diagonal estimates we prove a Nash type inequality for the related Dirichlet form. For the off-diagonal estimates we assume that the characteristic function of a Lévy (type) process is analytic, which allows to apply the complex analysis technique.

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

“…In the following lemma we improve the estimates obtained previously in [30,Lemma 2]. We use here in essential way the results of [21].…”

“…and, therefore, by (32), for every r > 0 and t > 0 the measuresP r t are absolutely continuous with respect to the Lebesgue measure with densitiesp r t ∈ C 1 b (R d ). We have P t =P r t * P r t , and p t =p r t * P r t , t > 0, We will estimate first the densitiesp r t ∈ C 1 b (R d ) using Lemma 4.2 and Theorem 6 of [21]. Let…”

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

“…In particular, for the process Y the estimate (1.5) holds with δ = 0. We note that equivalence between Nash inequalities of the form (1.4) and transition density estimates of the corresponding process is also considered in [18]. Some more general Nash inequalities were studied in [4,5,13].…”

Transition density estimates for diagonal systems of SDEs driven by cylindrical alpha-stable processes

“…Then, if (22) is true, then (20) holds for n + 1. Hence in order to complete the proof it is enough to show (22). To this end we consider 3 cases.…”

“…The problem of estimates of transition densities for jump Lévy and Lévy-type processes has been intensively studied in recent years see e.g. [11,19,2,10,7,22,16,20,28,21]. However, relatively few results concern processes with jump kernels which are not comparable to isotropic ones.…”