On some smoothening effects of the transition semigroup of a Lévy process

Let (Pt) be the transition semigroup of a Lévy process (Lt) taking values in a Hilbert space H. Let ν and N˜ respectively be the Lévy measure and compensated Poisson random measure of (Lt). It is shown that for any bounded and measurable function f,AqPtf(x)=1tE[f(Ltx)∫0t∫Hq(y)N˜(ds,dy)]for all t>0,x∈H, where Aq is some nonlocal operator. As a corollary,∫H|Ptf(x+y)−Ptf(x)|2ν(dy)≤1tPtf2(x)for all t>0,x∈H. As ν can be infinite this formula establishes some smoothening effect of the semigroup (Pt). In the paper some applications of the formula will be presented as well.