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

Abstract

Let \(\mathcal {X}\) be a separable Hilbert space with norm \(\left \|{\cdot }\right \|\) and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on \(\mathcal {X}\), let \(F:\mathcal {X}\rightarrow \mathcal {X}\) be a (smooth enough) function and let W(t) be a \(\mathcal {X}\)-valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator \(A:=-(1/2)Q^{2\alpha -1}:Q^{1-2\alpha }(\mathcal {X})\subseteq \mathcal {X}\rightarrow \mathcal {X}\). We are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation $$ \left\{\begin{array}{ll} dX(t,x)=\left( AX(t,x)+F(X(t,x))\right)dt+ Q^{\alpha}dW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. $$ and in its associated transition semigroup $$ P(t)\varphi(x):=\mathbb{E}[\varphi(X(t,x))], \qquad \varphi\in B_{b}(\mathcal{X}),\ t\in[0,T],\ x\in \mathcal{X}; $$ where \(B_{b}(\mathcal {X})\) 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 \(\mathcal {X}\). More precisely there exists K := K(F, T) > 0 such that for every \(\varphi \in B_{b}(\mathcal {X})\), \(x\in \mathcal {X}\), t ∈ (0, T] and \(h\in Q^{\alpha }(\mathcal {X})\) it holds $$ |P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-\alpha}h\|. $$