Heat kernels of non-symmetric jump processes: beyond the stable case

Abstract

Let $J$ be the Lévy density of a symmetric Lévy process in ${\mathbb{R}}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator $${\mathcal{L}}^{\kappa }f(x):=\underset{\epsilon \downarrow 0}{\lim }{\int }_{\{z\in {\mathbb{R}}^d:|z|>\epsilon \}}(f(x+z)-f(x))\kappa (x,z)J(z)dz,$$ where $\kappa (x,z)$ is a Borel measurable function on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ satisfying $0<{\kappa }_0\le \kappa (x,z)\le {\kappa }_1$, $\kappa (x,z)=\kappa (x,-z)$ and $|\kappa (x,z)-\kappa (y,z)|\le {\kappa }_2|x-y{|}^{\beta }$ for some $\beta \in (0,1)$. We construct the heat kernel $p^{\kappa }(t,x,y)$ of ${\mathcal{L}}^{\kappa }$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^{\kappa }$.

Gradient estimate improved, several errors corrected; 57 pages

References in corpus (2)

• Estimates of Dirichlet heat kernel for symmetric Markov processes
• Global Dirichlet Heat Kernel Estimates for Symmetric Lévy Processes in Half-space

Cited by in corpus (3)

• Heat Kernels for Non-symmetric Non-local Operators
• Heat kernel estimates for symmetric jump processes with mixed polynomial growths
• Williams decomposition for superprocesses