希尔伯特空间中（非高斯）转移半群的正则化性质
Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces
AI总结：
本文研究希尔伯特空间中半线性随机偏微分方程转移半群的正则化性质，证明在适当条件下，转移半群沿连续嵌入子空间满足关于时间与空间方向的定量Lipschitz型估计。
摘要：
设 $\mathcal{X}$ 为具有范数 $\\|\cdot\\|$ 的可分希尔伯特空间，并设 $T>0$。设 $Q$ 为 $\mathcal{X}$ 上的线性、自伴、正、迹类算子，设 $F:\mathcal{X}\rightarrow \mathcal{X}$ 为（足够光滑的）函数，并设 $W(t)$ 为 $\mathcal{X}$ 值柱面维纳过程。对于 $α\in [0,1/2]$，我们考虑算子 $A:=-(1/2)Q^{2α-1}:Q^{1-2α}(\mathcal{X})\subseteq \mathcal{X}\rightarrow \mathcal{X}$。我们关注半线性随机偏微分方程 \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*} 的温和解 $X(t,x)$ 及其关联的转移半群 \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*} 其中 $B_b(\mathcal{X})$ 是有界且博雷尔可测函数空间。我们将证明，在对 $Q$ 和 $F$ 的适当假设下，$P(t)$ 沿 $\mathcal{X}$ 的一个连续嵌入子空间具有正则化性质。更精确地说，存在 $K:=K(F,T)>0$，使得对每个 $φ\in B_b(\mathcal{X})$、$x\in \mathcal{X}$、$t\in(0,T]$ 和 $h\in Q^α(\mathcal{X})$，成立 \\[|P(t)φ(x+h)-P(t)φ(x)|\leq Kt^{-1/2}\\|Q^{-α}h\\|.\\]
英文摘要：
Let $\mathcal{X}$ be a separable Hilbert space with norm $\|\cdot\|$ 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 $α\in [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2α-1}:Q^{1-2α}(\mathcal{X})\subseteq \mathcal{X}\rightarrow \mathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation \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 $φ\in B_b(\mathcal{X})$, $x\in \mathcal{X}$, $t\in(0,T]$ and $h\in Q^α(\mathcal{X})$ it holds \[|P(t)φ(x+h)-P(t)φ(x)|\leq Kt^{-1/2}\|Q^{-α}h\|.\]
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
