Transition density estimates for diagonal systems of SDEs driven by cylindrical $\alpha$-stable processes

Abstract

We consider the system of stochastic differential equation $d{X}_t=A(X_{t-})d{Z}_t$, $X_0=x$, driven by cylindrical $\alpha$-stable process $Z_t$ in ${\mathbb{R}}^d$. We assume that $A(x)=(a_{ij}(x))$ is diagonal and $a_{ii}(x)$ are bounded away from zero, from infinity and Hölder continuous. We construct transition density $p^A(t,x,y)$ of the process $X_t$ and show sharp two-sided estimates of this density. We also prove Hölder and gradient estimates of $x\to p^A(t,x,y)$. Our approach is based on the method developed by Chen and Zhang.