Heat kernel estimates and their stabilities for symmetric jump processes with general mixed polynomial growths on metric measure spaces
Abstract
In this paper, we consider a symmetric pure jump Markov process $X$ on a general metric measure space that satisfies volume doubling conditions. We study estimates of the transition density $p(t,x,y)$ of $X$ and their stabilities when the jumping kernel for $X$ has general mixed polynomial growths. Unlike [24], in our setting, the rate function which gives growth of jumps of $X$ may not be comparable to the scale function which provides the borderline for $p(t,x,y)$ to have either neardiagonal estimates or offdiagonal estimates. Under the assumption that the lower scaling index of scale function is strictly bigger than $1$, we establish stabilities of heat kernel estimates. If underlying metric measure space admits a conservative diffusion process which has a transition density satisfying a general subGaussian bounds, we obtain heat kernel estimates which generalize [2, Theorems 1.2 and 1.4]. In this case, scale function is explicitly given by the rate function and the function $F$ related to walk dimension of underlying space. As an application, we proved that the finite moment condition in terms of $F$ on such symmetric Markov process is equivalent to a generalized version of Khintchinetype law of iterated logarithm at the infinity.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.10189
 Bibcode:
 2019arXiv190410189B
 Keywords:

 Mathematics  Probability;
 60J35;
 60J75;
 60F99