Heat kernel estimates for jump processes of mixed types on metric measure spaces

Abstract

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensitywhere ν is a probability measure on [α 1 , α 2 ] ⊂ (0, 2), c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c 0 (x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ 1 and γ 2 , where either γ 2 ≥ γ 1 > 0 or γ 1 = γ 2 = 0. This example contains mixed symmetric stable processes on R n as well as mixed relativistic symmetric stable Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday.

A geometric interpretation of the transition density of a symmetric Lévy process

“…Chen and T. Kumagai [11,12], A. Grigor'yan and J. Hu [19], and A. Grigor'yan, J. Hu and K.-S. Lau [20,21], to mention some of this work. Note the difference to our point of view.…”

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

“…Relativistic α-stable processes received intensive study in recent years and various fine properties of these processes have been obtained in [3,4,15,16,21,24,25,27,30]. In 2002, Ryznar [30] proved that, when D is a bounded C 1,1 domain in R d , the Green function G m D of the process X m in D is comparable to the Green function G D of the symmetric α-stable process X.…”

Jump Processes on Spaces With Non-Uniform Volume Growth

“…We generally observe a non-polynomial bound in Theorem 2.2(ii), and choosing a coarser polynomial bound would lead to an error exceeding the logarithmic scale. This is different from the case of uniform volume growth [16]. The first result is a standard upper bound for mean exit times prescribed by the jump measure.…”

“…As already noted in [16], the uniform doubling (1) is equivalent to saying that there exists some c > 0 and σ > 0 such that…”

“…We survey appropriate versions of some arguments in [16], [18] and [36]. The next is a Nash inequality in the form used in [16] and [36].…”