Analytic Properties of Markov Semigroup Generated by Stochastic Differential Equations Driven by L,vy Processes

We consider the stochastic differential equation (SDE) of the form where is globally Lipschitz continuous and L={L(t):ta0} is a L,vy process. Under this condition on sigma it is well known that the above problem has a unique solution X. Let be the Markovian semigroup associated to X defined by , ta0, , . Let B be a pseudo-differential operator characterized by its symbol q. Fix . In this article we investigate under which conditions on sigma, L and q there exist two constants gamma > 0 and C > 0 such that vertical bar BPtu vertical bar H-2(rho) <= C t(-gamma) vertical bar u vertical bar H-2(rho), (sic)u is an element of H-2(rho) (R-d), t > 0.