论文标题
无本地马尔可夫对称形式在无限的维空间上
Non-local Markovian symmetric forms on infinite dimensional spaces
论文作者
论文摘要
关于概率空间$（s，{\ cal b}（s），μ）$的非本地马尔可夫对称形式的一般定理，带有$ s $ s $ fr {é} chet空间，使得$ s \ s \ s \ subseT {引入了$ s $的$σ$ - $ s $，而$μ$是$ s $的borel概率度量。首先，一个非本地马尔可夫对称形式的家族$ {\ cal e} _ {（α）} $，$ 0 <α<2 $，在每个给定的$ l^2（s;μ）$中作用，定义为索引$α$表征非局限性的订单。然后，显示出所有表格$ {\ cal e} _ {（α）} $在$ \ bigCup_ {n \ in {\ Mathbb n}} c^{\ infty} _0（{\ Mathbb r}^n）$中$ l^2（s;μ）$可封闭。此外，给出了足够的条件，在这些条件下，封闭形式的封闭形式（是迪利奇形式）被严格构成准确的条件。最后，给出了与Dirichlet形式正确关联的狩猎过程的存在定理。通过这些狩猎过程，将上述定理应用于欧几里得$φ^4_d $字段的随机量化问题，以$ d = 2，3 $。
General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces $(S, {\cal B}(S), μ)$, with $S$ Fr{é}chet spaces such that $S \subset {\mathbb R}^{\mathbb N}$, ${\cal B}(S)$ is the Borel $σ$-field of $S$, and $μ$ is a Borel probability measure on $S$, are introduced. Firstly, a family of non-local Markovian symmetric forms ${\cal E}_{(α)}$, $0 < α< 2$, acting in each given $L^2(S; μ)$ is defined, the index $α$ characterizing the order of the non-locality. Then, it is shown that all the forms ${\cal E}_{(α)}$ defined on $\bigcup_{n \in {\mathbb N}} C^{\infty}_0({\mathbb R}^n)$ are closable in $L^2(S;μ)$. Moreover, sufficient conditions under which the closure of the closable forms, that are Dirichlet forms, become strictly quasi-regular, are given. Finally, an existence theorem for Hunt processes properly associated to the Dirichlet forms is given. The application of the above theorems to the problem of stochastic quantizations of Euclidean $Φ^4_d$ fields, for $d =2, 3$, by means of these Hunt processes is indicated.
