Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces I. The closability and quasi-regularity

General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces (S, ℬ(S), μ ) , with S Fréchet spaces such that S ⊂ℝ^ℕ , 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 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 ℰ_(α ) defined on ⋃ _n ∈ℕ C^∞_0(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 ^4_d fields, for d =2, 3 , by means of these Hunt processes is indicated.