Regularizing Properties of (Non-Gaussian) Transition Semigroups in Hilbert Spaces

Let X be a separable Hilbert space with norm ∥ ⋅ ∥ and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on X, let F: X→ X be a (smooth enough) function and let W(t) be a X-valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator A: = − (1 / 2) Q2α−1: Q1−2α(X) ⊆ X→ X. We are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation{dX(t,x)=(AX(t,x)+F(X(t,x)))dt+QαdW(t),t∈(0,T];X(0,x)=x∈X, and in its associated transition semigroupP(t)φ(x):=E[φ(X(t,x))],φ∈Bb(X),t∈[0,T],x∈X; where Bb(X) is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on Q and F, P(t) enjoys regularizing properties, along a continuously embedded subspace of X. More precisely there exists K := K(F, T) > 0 such that for every φ∈ Bb(X) , x∈ X, t ∈ (0, T] and h∈ Qα(X) it holds| P(t) φ(x+ h) − P(t) φ(x) | ≤ Kt− 1 / 2∥ Q−αh∥.