Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent Lévy processes

Let $(P_t)$ be the transition semigroup of the Markov family $(X^x(t))$ defined by SDE $d X= b(X) dt + d Z, \qquad X(0)=x,$ where $Z=\left(Z_1, \ldots, Z_d\right)^*$ is a system of independent real-valued Lévy processes. Using the Malliavin calculus we establish the following gradient formula $\nabla P_tf(x)= \mathbb{E}\, f\left(X^x(t)\right) Y(t,x), \qquad f\in B_b(\mathbb{R}^d),$ where the random field $Y$ does not depend on $f$. Sharp estimates on $\nabla P_tf(x)$ when $Z_1, \ldots , Z_d$ are $α$-stable processes, $α\in (0,2)$, are also given.

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