Abstract
General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces (S, mathcal{B}(S), mu ), with S Fréchet spaces such that S subset {{mathbb {R}}}^{{mathbb {N}}}, {{mathcal {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 {{mathcal {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 {{mathcal {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 varPhi ^4_d fields, for d =2, 3, by means of these Hunt processes is indicated.
Highlights
On the real L2(S; μ) space, for each 0 < α < 2, we give an explicit formulation of α-stable type non-local quasi-regular Dirichlet forms (E(α), D(E(α))) (with a domain D(E(α))), and show the existence of S-valued Hunt processes properly associated to (E(α), D(E(α))). α-stable is understood in analogy with the α-stable Dirichlet forms defined on L2(Rd ), for d ∈ N, e.g., in [49], Sect. 5
In the literature quoted above concerning local Dirichlet forms we find several applications to stochastic quantizations, where the Markov processes constructed are diffusion processes associated to local Dirichlet forms
We introduce below a standard procedure of application of Theorems 1, 2, 3 and 4 to the problem of stochastic quantizations of Euclidean quantum fields, by means of the Hunt processes in Theorem 4
Summary
We consider a space S that is either a real Banach space l p, 1 ≤ p ≤ ∞, with suitable weights, or the direct product space RN (with R and N the spaces of real numbers and natural numbers, respectively). The stochastic quantizations are realized by Hunt processes properly associated to the non-local Dirichlet forms (E(α), D(E(α))) for 0 < α ≤ 1 In these examples, in order to apply Theorem 4 to the Euclidean free field measure and to the Φ24 and Φ34 field measures, respectively, on the Schwartz space of tempered distributions, we firstly certify that these measures have support in the Hilbert spaces H−2 and H−3 (cf (5.11)), respectively, and define the isometric isomorphism τ−2 : H−2 → a weighted l2 space , l(2λ4) (c f.(5.27)) i and, τ−3 : H−3 → a weighted l2space, l(2λ6) (c f.(5.56)), i and we identify H−2 with l(2λ4) and H−3 with l(2λ6), respectively.
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