Heat kernel estimates for non-symmetric stable-like processes

Abstract

Let $d\ge 1$ and $0<\alpha <2$. Consider the integro-differential operator \[ \mathcal{L}f(x) =\int_{\mathbb{R}^{d}\backslash\{0\}}\left[f(x+h)-f(x)-χ_α(h)\nabla f(x)\cdot h\right]\frac{n(x,h)}{|h|^{d+α}}\mathrm{d}h+\mathbf{1}_{α>1}b(x)\cdot\nabla f(x), \] where ${\chi }_{\alpha }(h):=1_{\alpha >1}+1_{\alpha =1}{1}_{\{|h|\le 1\}}$, $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$ is bounded measurable, and $n:{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R}$ is measurable and bounded above and below respectively by two positive constants. Further, we assume that $n(x,h)$ is Hölder continuous in $x$, uniformly with respect to $h\in {\mathbb{R}}^d$. In the case $\alpha =1,$ we assume additionally ${\int }_{\partial B_r}n(x,h)h\mathrm{d}{S}_r(h)=0$, $\forall r\in (0,\infty )$, where $\mathrm{d}{S}_r$ is the surface measure on $\partial B_r$, the boundary of the ball with radius $r$ and center $0$. In this paper, we establish two-sided estimates for the heat kernel of the Markov process associated with the operator $\mathcal{L}$. This extends a recent result of Z.-Q. Chen and X. Zhang.

37 pages

Cited by in corpus (3)

• Heat kernels for time-dependent non-symmetric stable-like operators
• Heat kernels of non-symmetric jump processes with exponentially decaying jumping kernel
• Uniform dimension results for a family of Markov processes