论文标题
（非高斯）在希尔伯特空间中的（非高斯）过渡半群的正规化特性
Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces
论文作者
论文摘要
令$ \ mathcal {x} $为可分离的希尔伯特空间，具有norm $ \ | \ cdot \ | $，让$ 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.对于$α\在[0,1/2] $中，我们考虑运算符$ a：= - （1/2）q^{2α-1}：q^{1-2α}（\ Mathcal {x}）\ subseteq \ subseteq \ Mathcal {x}}我们对半连接随机部分微分方程的温和解决方案$ x（t，x）$感兴趣\ begin {chatch*} \ left \ oben {arnay} {array} {ll} {ll} dx（t，x）= \ ax ax（ax（t，x）+ f（x）+ f（x（x（x）\ big） t \ in（0，t]; \\ x（0，x）= x \ in \ Mathcal {x}，\ end {array} \ right。 φ\在b_b（\ Mathcal {x}）中，\ t \ in [0，t]，\ x \ in \ Mathcal {x}; $ p（t）$享受正规化的属性，沿着$ \ mathcal {x} $的连续嵌入子空间。 $ h \ in q^α（\ mathcal {x}）$它保持\ [| p（t）φ（x+h）-p（t）φ（x）| \ leq kt kt^{ - 1/2} \ | q^{ - | q^{ - α}
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 \[|P(t)φ(x+h)-P(t)φ(x)|\leq Kt^{-1/2}\|Q^{-α}h\|.\]
