论文标题
关于与半连续随机偏微分方程相关的过渡半群的发电机
On generators of transition semigroups associated to semilinear stochastic partial differential equations
论文作者
论文摘要
令$ \ Mathcal {x} $为真正可分离的希尔伯特空间。令$ q $为$ \ MATHCAL {x} $上的线性，自我接合，积极，跟踪类操作员，让$ f：\ nathcal {x} \ rightArrow \ Mathcal \ Mathcal {x} $成为（足够平滑）功能，让$ \ \ {w（w（t）圆柱状维纳过程。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}}我们对半连续的随机部分偏微分方程的温和解决方案$ x（t，x）$感兴趣\ begin {chater} \ left \ {\ oken {array} {ll} {ll} dx（t，x）= \ ax（x） x（0，x）= x \ in \ mathcal {x}，\ end {array} \ right。 \ end {chater}及其相关的过渡semigroup \ begin {align} p（t）φ（x）：= e [φ（x（x（t，x））] \ end {align}其中$ b_b（\ mathcal {x}）$是$ \ Mathcal {x} $上的实价，有限和borel可测量的函数的空间。在本文中，我们研究了Space $ l^2（\ Mathcal {X}，ν）$的Semigroup $ P（T）$的行为，其中$ν$是\ eqref {tropical}的唯一不变概率度量，当$ f $是消散性的，并且具有多项态度的增长。然后，我们证明了对数Sobolev和Poincaré不平等，并研究了固定方程式的最大sobolev规律性\ [λu-n_2 u = f，\ qquadλ> 0，\ f \ f \ in l^2（\ natcal {x}}，x}，n是$ n_2 $ l^2（\ Mathcal {x}，ν）$。
Let $\mathcal{X}$ be a real separable Hilbert space. 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 $\{W(t)\}_{t\geq 0}$ 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>0;\\ X(0,x)=x\in \mathcal{X}, \end{array} \right. \end{gather} and in its associated transition semigroup \begin{align} P(t)φ(x):=E[φ(X(t,x))], \qquad φ\in B_b(\mathcal{X}),\ t\geq 0,\ x\in \mathcal{X}; \end{align} where $B_b(\mathcal{X})$ is the space of the real-valued, bounded and Borel measurable functions on $\mathcal{X}$. In this paper we study the behavior of the semigroup $P(t)$ in the space $L^2(\mathcal{X},ν)$, where $ν$ is the unique invariant probability measure of \eqref{Tropical}, when $F$ is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincaré inequalities and we study the maximal Sobolev regularity for the stationary equation \[λu-N_2 u=f,\qquad λ>0,\ f\in L^2(\mathcal{X},ν);\] where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(\mathcal{X},ν)$.
