Heat kernels for non-symmetric diffusion operators with jumps

TL;DR

This paper proves the existence, uniqueness, and sharp estimates of heat kernels for a broad class of non-symmetric diffusion operators with jumps, combining analytic and probabilistic methods.

Contribution

It introduces new results on heat kernels for time-dependent non-symmetric operators with jumps, including sharp estimates and a unified analytic-probabilistic approach.

Findings

Existence and uniqueness of heat kernels established.

Sharp two-sided and gradient estimates obtained.

Applicable to time-dependent non-symmetric operators with jumps.

Abstract

For $d\geq 2$, we establish the existence and uniqueness of heat kernels for a large class of time-dependent second order diffusion operator with jumps, which is the sum of time-dependent of a second order elliptic differential operators non-divergence form and a non-local $\alpha$-stable-type operator with bounded time-dependent coefficient. Moreover, we obtain sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernels under some mild conditions. Our approach is mainly analytic but also uses some probabilistic techniques.