Non-local Markovian symmetric forms on infinite dimensional spaces

Abstract

General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces $(S, {\cal B}(S), \mu)$, with $S$ Fr{\'e}chet spaces such that $S \subset {\mathbb R}^{\mathbb N}$, ${\cal B}(S)$ is the Borel $\sigma$-field of $S$, and $\mu$ is a Borel probability measure on $S$, are introduced. Firstly, a family of non-local Markovian symmetric forms ${\cal E}_{(\alpha)}$, $0 < \alpha < 2$, acting in each given $L^2(S; \mu)$ is defined, the index $\alpha$ characterizing the order of the non-locality. Then, it is shown that all the forms ${\cal E}_{(\alpha)}$ defined on $\bigcup_{n \in {\mathbb N}} C^{\infty}_0({\mathbb R}^n)$ are closable in $L^2(S;\mu)$. 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 $\Phi^4_d$ fields, for $d =2, 3$, by means of these Hunt processes is indicated.