Heat kernel bounds for a large class of Markov process with singular jump

Abstract

Let $Z=(Z^1,\dots ,Z^d)$ be the $d$-dimensional Lévy processes where $Z^i$'s are independent $1$-dimensional Lévy processes with jump kernel $J^{\varphi ,1}(u,w)=|u-w{|}^{-1}\varphi (|u-w|{)}^{-1}$ for $u,w\in \mathbb{R}$. Here $\varphi$ is an increasing function with weak scaling condition of order $\underline{\alpha },\overline{\alpha }\in (0,2)$. Let $J(x,y)\asymp J^{\varphi }(x,y)$ be the symmetric measurable function where \begin{align*} J^ϕ(x,y):=\begin{cases} J^{ϕ, 1}(x^i, y^i)\qquad&\text{ if $x^i\ne y^i$ for some $i$ and $x^j=y^j$ for all $j\ne i$}\\ 0\qquad&\text{ if $x^i\ne y^i$ for more than one index $i$.} \end{cases} \end{align*} Corresponding to the jump kernel $J$, we show the existence of non-isotropic Markov processes $X:=(X^1,\dots ,X^d)$ and obtain sharp two-sided heat kernel estimates for the transition density functions.