Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces

Abstract

Let $\mathcal{X}$ be a separable Hilbert space with norm $\Vert \cdot \Vert$ and let $T>0$. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $\mathcal{X}$, let $F:\mathcal{X}\to \mathcal{X}$ be a (smooth enough) function and let $W(t)$ be a $\mathcal{X}$-valued cylindrical Wiener process. For $\alpha \in [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2\alpha -1}:Q^{1-2\alpha }(\mathcal{X})\subseteq \mathcal{X}\to \mathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation \begin{gather*} \left\{\begin{array}{ll} dX(t,x)=\big(AX(t,x)+F(X(t,x))\big)dt+ Q^αdW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. \end{gather*} and in its associated transition semigroup \begin{align*} P(t)φ(x):=\mathbb{E}[φ(X(t,x))], \qquad φ\in B_b(\mathcal{X}),\ t\in[0,T],\ x\in \mathcal{X}; \end{align*} 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 $\phi \in B_b(\mathcal{X})$, $x\in \mathcal{X}$, $t\in (0,T]$ and $h\in Q^{\alpha }(\mathcal{X})$ it holds \[|P(t)φ(x+h)-P(t)φ(x)|\leq Kt^{-1/2}\|Q^{-α}h\|.\]

References in corpus (1)

• Formulae for the derivatives of heat semigroups

Cited by in corpus (5)

• Schauder regularity results in separable Hilbert spaces
• Differentiability in infinite dimension and the Malliavin calculus
• Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise
• Regarding the domain of non-symmetric and, possibly, degenerate Ornstein--Uhlenbeck operators in separable Banach spaces
• $L^2$-theory for transitions semigroups associated to dissipative systems