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Non-Local Markovian Symmetric Forms on Infinite Dimensional Spaces

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Non-Local Markovian Symmetric Forms on Infinite Dimensional Spaces Non-local Markovian symmetric forms on infinite dimensional spaces Part 2. Examples: non local stochastic quantization of space cut-off quantum fields and infinite particle systems Sergio Albeverio ,∗ Toshinao Kagawa ,† Shyuji Kawaski ,‡ Yumi Yahagi ,§ and Minoru W. Yoshida ¶ May 13, 2021 Abstract The general framework on the non-local Markovian symmetric forms on weighted lp (p [1, ]) spaces constructed by [A,Kagawa,Yahagi,Y 2020], by restricting the ∈ ∞ situation where p = 2, is applied to such measure spaces as the space cut-off P (φ)2 Euclidean quantum field, the 2-dimensional Euclidean quantum fields with exponential and trigonometric potentials, and the field describing a system of an infinite number of classical particles. For each measure space, the Markov process corresponding to the non-local type stochastic quantization is constructed. keywords: Non local Dirichlet forms on infinite dimensional spaces, space cut-off P (φ)2, exp φ, sin φ-quantum field models, Euclidean quantum field, infinite particle systems, non-local stochastic quantization. MSC (2020): 31C25, 46E27, 46N30, 46N50, 47D07, 60H15, 60J46, 60J75, 81S20 arXiv:2105.05593v1 [math-ph] 12 May 2021 1 Introduction and preliminaries In this paper we apply a general framework on the non-local Markovian symmetric forms on weighted lp (p [1, ]) spaces constructed by [A,Kagawa,Yahagi,Y 2020], by restrict- ing the situation where∈ ∞p = 2, i.e., the weighted l2-space, to the stochastic quantizations ∗Inst. Angewandte Mathematik, and HCM, Univ. Bonn, Germany, email :[email protected] †Dept. Information Systems Kanagawa Univ., Yokohama, Japan ‡Dept. Mathematical Sciences and Phyisics, Iwate Univ., Morioka, Japan §Dept. Mathematical information Tokyo Univ. of Information, Chiba, Japan ¶Dept. Information Systems Kanagawa Univ., Yokohama, Japan, email: [email protected] 1 of the space cut-off P (φ)2 Euclidean quantum field, the 2-dimensional Euclidean quantum fields with exponential and trigonometric potentials, and a field of classical (infinite) particle systems. Then, for each random field, the Markov process corresponding to the non-local type stochastic quantization is constructed. As far as we know, there exists no consider- ations on the non-local type stochastic quantizations for such random fields through the arguments by the Dirichlet forms, that are non-local (cf., [A 2003], [A 2016], [A,Ma,R 2015], [A,DeV,Gu1,2], [A,Di Persio, Mastrogiacomo, Smii 2016], [A,H-K 76], [A,H-K 77], [A,Kusuoka-sei 2017], [A,R 89], [A,R 90], [A,R 91], [A,R¨udiger 2003], [Cat,Chouk 2018], [Da Prato, Debussche 03], [Gu,Ho 2019], [Hairer 2014], [Hairer,Mattingly 2016], [Mourrat,Weber 2017], and also for a historical aper¸cu on the stochastic quantizations of several random fields cf. [A,Kagawa,Yahagi,Y 2020] and references therein). In this section, we first recall the abstract results on the non-local Dirichlet forms defined on the Fr´echet spaces provided in [A,Kagawa,Yahagi,Y 2020], and its application to the stochastic quantization of the Euclidean free quantum field which also has been considered in [A,Kagawa,Yahagi,Y 2020]. By these preparations, in the next section we proceed to construct the solutions of stochastic quantizations corresponding to the space cut-off P (φ)2 Euclidean quantum field, the 2-dimensional Euclidean fields with exponential and trigono- metric potentials, and a field of classical (infinite) particle systems. Here, we limit ourselves to recalling the results in [A,Kagawa,Yahagi,Y 2020] that will be applied to the stochastic quantizations mentioned above. Precisely, for the applications, we restrict our selves to the formulations on the weighted l2 spaces and the non-local Dirichlet forms with the index 0 <α 1, the index characterizing the order of the non-locality which has a corresponcence to the≤ index of the α stable processes. The abstract state spaces S, on which we define the Markovian symmetric forms, are the 2 2 weighted l spaces, denoted by l , with a given weight (βi)i∈N, βi 0, i N, such that (βi) ≥ ∈ ∞ 2 N 2 1 R 2 2 S = l(βi) x =(x1, x2,... ) : x l ( βi xi ) < . (1.1) ≡ ∈ k k (βi) ≡ | | ∞ i=1 X We denote by (S) the Borel σ-field of S. Suppose that we are given a Borel probability B measure µ on (S, (S)). For each i N, let σic be the sub σ-field of (S) that is generated by the Borel sets B ∈ B B = x S x B ,...x B , j = i, B 1, k =1,...,n, n N, (1.2) ∈ j1 ∈ 1 jn ∈ n k 6 k ∈ B ∈ 1 1 where denotes the Borel σ-field of R . Thus, σic is the smallest σ-field that includes every B B given by (1.2). Namely, σ c is the sub σ-field of (S) generated by the variables x x , i.e., i B \ i the variables except of the i-th variable xi. For each i N, let µ( σic ) be the conditional probability, a one-dimensional probability distribution∈ (i.e., a probabilit· y distribution for the i-th component xi) valued σic measurable function, that is characterized by (cf. (2.4) of [A,R 91]) 1 µ x : x A B = µ(A σ c ) µ(dx), A , B σ c . (1.3) { i ∈ } ∩ i ∀ ∈ B ∀ ∈ i ZB 2 Define 1 2 2 2 L (S; µ) f f : S R, measurable and f L2 = f(x) µ(dx) < , (1.4) ≡ → k k S | | ∞ Z and ∞ ∞ n 2 C = f(x ,...,x ) IR(x ) f C (R R), n N L (S; µ), (1.5) F 0 1 n · i ∃ ∈ 0 → ∈ ⊂ n i≥1 o Y ∞ n n where C0 (R R) denotes the space of real valued infinitely differentiable functions on R with compact supports.→ 2 On L (S; µ), for any 0 <α 1, let us define the Markovian symmetric form (α) called individually adapted Markovian≤ symmetric form of index α to the measure µ, theE definition of which is a natural analogue of the one for α-stable type (non local) Dirichlet form on Rd, d< (cf. (5.3), (1.4) of [Fukushima,Uemura 2012], also cf. Remark 1.1 below for the corresponding∞ local Dirichlet forms on L2(Rn; µ) with finite n N). ∈ For each 0 <α 1 and i N, and for the variables ≤ ∈ y , y′ R1, y = y′, x =(x ,...,x , x , x ,... ) S and x x (x ,...,x , x ,... ), i i ∈ i 6 i 1 i−1 i i+1 ∈ \ i ≡ 1 i−1 i+1 let Φ (u, v; y ,y′, x x ) α i i \ i 1 u(x ,...,x ,y , x ,... ) u(x ,...,x ,y′, x ,... ) ≡ y y′ α+1 × 1 i−1 i i+1 − 1 i−1 i i+1 | i − i| v(x ,...,x ,y , x ,... ) v(x ,...,x ,y′, x ,... ) , (1.6) × 1 i−1 i i+1 − 1 i−1 i i+1 then define (i) (u, v) I (y ) Φ (u, v; y , x , x x ) µ dy σ c µ(dx), (1.7) E(α) ≡ {y : y6=xi} i α i i \ i i i ZS ZR n o for any u, v such that the right hand side of (1.7) is finite, where for a set A and a variable y, IA(y) denotes the indicator function, and in the sequel, to simplify the notations, we denote I{y : y6=xi}(yi) by, e.g., I{yi6=xi}(yi) or I{yi6=xi}. By , we denote the subset of the space of real valued (S)-measurable functions such Di B that the right hand side of (1.7) is finite for any u, v . Let us call ( (i) , ) this form, ∈ Di E(α) Di being its domain. Then define Di (u, v) (i) (u, v), u, v . (1.8) E(α) ≡ E(α) ∀ ∈ Di i N i N X∈ \∈ It is easy to see that for the Lipschiz continuous functionsu ˜ C∞(Rn R) C∞ and ∈ 0 → ⊂F 0 v˜ C∞(Rm R) C∞, n, m N, which are representatives of u C∞ and v C∞ ∈ 0 → ⊂F 0 ∈ ∈F 0 ∈F 0 3 (i) respectively, n, m N, (α)(˜u, v˜) and (α)(˜u, v˜) are finite. Actually, in [A,Kagawa,Yahagi,Y ∈ E E ∞ ∞ 2020] it is proved that (1.7) and (1.8) are well defined for C0 , and hence C0 i∈N i, i.e., it is shown that for any real valued (S)-measurableF function u on SF, such⊂ that ∩ uD= B ∞ 0, µ a.e., it holds that (α)(u,u)=0; and for any u, v C0 , there corresponds only one value− (u, v) R.E Moreover, in [A,Kagawa,Yahagi,Y∈ F 2020] it is shown that is E(α) ∈ E(α) a closable Markovian symmetric form. Precisely, the following Theorem 1 holds, which is a restatement of a result given by [A,Kagawa,Yahagi,Y 2020] and shall be applied to the subsequent discussions in the present paper (in [A,Kagawa,Yahagi,Y 2020], not only for 0 < α 1 but also for 0 <α< 2, and for the state spaces S as weighted lp spaces, 1 p ≤ , those theorems including the statements corresponding to Theorems 1, 2 and 3 introduced≤ ≤∞ in this paper are provided): Theorem 1 (The closability) For the symmetric non-local forms (α), 0 <α 1 given by (1.8) the following hold: E ≤ ∞ i) (α) is well-defined on C0 ; E ∞ F 2 ii) ( (α), C0 ) is closable in L (S; µ); E F ∞ iii) ( (α), C0 ) is Markovian. E F ∞ Thus, for each 0 < α 1, the closed extension of ( (α), C0 ) denoted by ( (α), ( (α))) with the domain ( ≤), is a non-local Dirichlet formE on LF2(S; µ).
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