Strong Feller property for SDEs driven by multiplicative cylindrical stable noise

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 $A(x) = (a_{ij}(x))$isboundedawayfromzero,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.