General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces \( (S, {\cal B}(S), \mu) \), with \(S\) Fréchet spaces such that \(S \subset R^N \), \( {\cal B}(S) \) is the Borel \( \sigma\)-field of \(S\), and \(\mu\) is a Borel probability measure on \(S\), are introduced. Precisely, 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, we see that all the forms \({\cal E}_{(\alpha)}\) defined on \(\bigcup_{n \in N} C^{\infty}_0(R^n)\) 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.

Through these, an existence theorem for Hunt processes properly associated to the Dirichlet forms is given.

The application of the above general theorems we consider the problem of stochastic quantizations of Euclidean \(\Phi^4_d\) fields, for \(d =2, 3\), by means of these Hunt processes is indicated.

Also, some of the important corresponding problems, e.g., the stochastic mechanics, Malliavin calculus for jump processes, etc., shall be mentioned.