Strong Feller property for SDEs driven by multiplicative cylindrical stable noise

Abstract

We consider the stochastic differential equation $d{X}_t=A(X_{t-})d{Z}_t$, $X_0=x$, driven by cylindrical $\alpha$-stable process $Z_t$ in $R^d$, where $\alpha \in (0,1)$ and $d\ge 2$. We assume that the determinant of $A(x)=(a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. We show that for any fixed $\gamma \in (0,\alpha )$ the semigroup $P_t$ of the process $X_t$ satisfies $|P_{t}f(x)-P_{t}f(y)|\le c{t}^{-\gamma /\alpha }|x-y{|}^{\gamma }||f|{|}_{\infty }$ for arbitrary bounded Borel function $f$. Our approach is based on Levi's method.

We corrected the mistake in inequality (64)