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

Abstract

AbstractLet $$(P_t)$$ ( P t ) be the transition semigroup of the Markov family $$(X^x(t))$$ ( X x ( t ) ) defined by SDE $$\begin{aligned} \mathrm{d}X= b(X)\mathrm{d}t + \mathrm{d}Z, \qquad X(0)=x, \end{aligned}$$ d X = b ( X ) d t + d Z , X ( 0 ) = x , where $$Z=\left( Z_1, \ldots , Z_d\right) ^*$$ Z = Z 1 , … , Z d ∗ is a system of independent real-valued Lévy processes. Using the Malliavin calculus we establish the following gradient formula $$\begin{aligned} \nabla P_tf(x)= {\mathbb {E}}\, f\left( X^x(t)\right) Y(t,x), \qquad f\in B_b({\mathbb {R}}^d), \end{aligned}$$ ∇ P t f ( x ) = E f X x ( t ) Y ( t , x ) , f ∈ B b ( R d ) , where the random field Y does not depend on f. Moreover, in the important cylindrical $$\alpha $$ α -stable case $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) , where $$Z_1, \ldots , Z_d$$ Z 1 , … , Z d are $$\alpha $$ α -stable processes, we are able to prove sharp $$L^1$$ L 1 -estimates for Y(t, x). Uniform estimates on $$\nabla P_tf(x)$$ ∇ P t f ( x ) are also given.