## The Annals of Probability

### Heat kernel estimates for symmetric jump processes with mixed polynomial growths

#### Abstract

In this paper, we study the transition densities of pure-jump symmetric Markov processes in $\mathbb{R}^{d}$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of the transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Lévy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment condition on such symmetric Markov process is equivalent to the Khintchine-type law of iterated logarithm at infinity.

#### Article information

Source
Ann. Probab., Volume 47, Number 5 (2019), 2830-2868.

Dates
Received: May 2018
Revised: November 2018
First available in Project Euclid: 22 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1571731438

Digital Object Identifier
doi:10.1214/18-AOP1323

Mathematical Reviews number (MathSciNet)
MR4021238

#### Citation

Bae, Joohak; Kang, Jaehoon; Kim, Panki; Lee, Jaehun. Heat kernel estimates for symmetric jump processes with mixed polynomial growths. Ann. Probab. 47 (2019), no. 5, 2830--2868. doi:10.1214/18-AOP1323. https://projecteuclid.org/euclid.aop/1571731438

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