Regularizing Properties of (Non-Gaussian) Transition Semigroups in Hilbert Spaces
نویسندگان
چکیده
Let $\mathcal{X}$ be a separable Hilbert space with norm $\|\cdot\|$ and let $T>0$. $Q$ linear, self-adjoint, positive, trace class operator on $\mathcal{X}$, $F:\mathcal{X}\rightarrow \mathcal{X}$ (smooth enough) function $W(t)$ $\mathcal{X}$-valued cylindrical Wiener process. For $\alpha\in [0,1/2]$ we consider the $A:=-(1/2)Q^{2\alpha-1}:Q^{1-2\alpha}(\mathcal{X})\subseteq \mathcal{X}\rightarrow \mathcal{X}$. We are interested in mild solution $X(t,x)$ of 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^{\alpha}dW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. \end{gather*} its associated transition semigroup \begin{align*} 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}; \end{align*} where $B_b(\mathcal{X})$ is bounded Borel measurable functions. will show that under suitable hypotheses $F$, $P(t)$ enjoys regularizing properties, along continuously embedded subspace $\mathcal{X}$. More precisely there exists $K:=K(F,T)>0$ such for every $\varphi\in B_b(\mathcal{X})$, $x\in \mathcal{X}$, $t\in(0,T]$ $h\in Q^\alpha(\mathcal{X})$ it holds \[|P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-\alpha}h\|.\]
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ژورنال
عنوان ژورنال: Potential Analysis
سال: 2021
ISSN: ['1572-929X', '0926-2601']
DOI: https://doi.org/10.1007/s11118-021-09931-2