Non-Local Markovian Symmetric Forms on Infinite Dimensional Spaces

arXiv:2105.05593v1 [math-ph] 12 May 2021 nti ae eapyagnrlfaeoko h o-oa Marko non-local the on framework general a weighted apply on we paper preliminaries this In and Introduction 1 n h iuto where situation the ing at2 xmls o oa tcatcqatzto fspac of quantization stochastic local non Examples: 2. Part ¶ ∗ ‡ † § nt neadeMteai,adHM nv on emn,em Germany, Bonn, Univ. HCM, and Mathematik, Angewandte Inst. et ahmtclSine n hiis wt nv,Mroa Ja Morioka, Univ., Iwate Phyisics, and Sciences Japan Yokohama, Mathematical Univ., Dept. Kanagawa Systems Information Dept. et ahmtclifrainTkoUi.o nomto,Chiba Information, of Univ. Tokyo information Mathematical Dept. et nomto ytm aaaaUi. ooaa aa,em Japan, Yokohama, Univ., Kanagawa Systems Information Dept. exp S (2020) MSC quantization. stochastic lsia atce.Frec esr pc,teMro pro keywords Markov the space, measure non-local each sys For a describing particles. field classical the and potentials, quant trigonometric Euclidean and 2-dimensional the field, quantum Euclidean iuto where situation l o-oa akva ymti om ninfinite on forms symmetric Markovian Non-local p Sergio ( h eea rmwr ntennlclMroinsymmetric Markovian non-local the on framework general The φ p sin , ∈ l [1 p φ yesohsi uniaini constructed. is quantization stochastic type Albeverio qatmfil oes ulda unu ed nnt atcesyste particle infinite field, quantum Euclidean models, field -quantum , : ( ∞ p Yumi o oa iihe om nifiiedmninlsae,saecut-o space spaces, dimensional infinite on forms Dirichlet local Non )sae osrce y[,aaaYhg, 00,b rest by 2020], [A,Kagawa,Yahagi,Y by constructed spaces ]) ∈ : 12,4E7 63,4N0 70,6H5 04,6J5 81S 60J75, 60J46, 60H15, 47D07, 46N50, 46N30, 46E27, 31C25, [1 p , ,i ple osc esr pcsa h pc cut-off space the as spaces measure such to applied is 2, = ∞ Yahagi p )sae osrce y[,aaaYhg, 00,b restrict- by 2020], [A,Kagawa,Yahagi,Y by constructed spaces ]) ,ie,teweighted the i.e., 2, = Toshinao , ∗ iesoa spaces dimensional n Minoru and , § a 3 2021 13, May atcesystems particle Abstract 1 Kagawa l 2 sae otesohsi quantizations stochastic the to -space, u-ffqatmfilsadinfinite and fields quantum cut-off e .Yoshida W. Shyuji , † e fa nnt ubrof number infinite an of tem mfilswt exponential with fields um i:[email protected] ail: escrepnigt the to corresponding cess Japan , pan i :[email protected] ail om nweighted on forms insmercforms symmetric vian Kawaski s non-local ms, ¶ 20 itn the ricting ff P P ( ( φ φ ) , ‡ ) 2 2 , 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 considerations 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,Rüdiger 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 Fréchet 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 trigonometric 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). E D E D E(α)

Remark 1.1 In [A,Kagawa,Yahagi,Y 2020], the symmetric form (i) is considered for E(α) 0 <α< 2. Then, the non-local symmetric form (i) defined by (1.6) and (1.7), by E(α) extending the definition for 0 <α< 2, can be interpreted as non-local and local symmetric ∞ d forms on the finite dimensional linear space C0 (R R) (cf., e.g., Example 4 in section 1.2 of [Fukushima 80], and section II-2 of [M,R 92]), the→ space of real valued smooth functions with compact supports on Rd with some finite d N: For simplicity, let d =1 and for the Borel probability measure µ, suppose that there exists∈ a smooth bounded probability density function ρ (R R), an element of Schwartz space of rapidly decreasing functions, such ∈ S → that 0 < ρ(x) < , x R. Then (i) is interpreted as follows: ∞ ∀ ∈ E(α) (f(y) f(x))(g(y) g(x)) ∞ R R − 1+α − ρ(y) ρ(x) dydx, f,g C0 ( ). 2 y x ∈ → ZR \”diagonal set” | − | Next, for each 0 <α< 2, let 1 1 M(α) (1 α) 2−α , ≡ − 2 then, for f, g C∞(R R), it holds that ∈ 0 → ′ ′ ′ (f(y ) f(x))(g(y ) g(x)) ′ ′ lim I{y:y6=x}(y ) − − ρ(y ) dy ρ(x) dx, α↑2 { y′ x 1+α } ZR Z[x−M(α),x+M(α)] | − | = f ′(x) g′(x)(ρ(x))2 dx. ZR 4 Also, for each 0 <α< 2 and each x R, if we let ∈ 1 1 M(α; x) ρ(x)−1(1 α) 2−α , ≡ − 2 then, for f, g C∞(R R), it holds that ∈ 0 → ′ ′ ′ (f(y ) f(x))(g(y ) g(x)) ′ ′ lim I{y:y6=x}(y ) − − ρ(y ) dy ρ(x)dx, α↑2 { y′ x 1+α } ZR Z[x−M(α;x),x+M(α;x)] | − | = f ′(x) g′(x) ρ(x) dx. ZR Considerations for the infinite dimensional situation corresponding to the above finite dimensional observation will be carried out in forthcoming work. The following theorem is also a part of the main results provided by [A,Kagawa,Yahagi,Y 2020] on the sufficient conditions (cf. Theorem 2 and Theorem 3 below) under which the Dirichlet forms (i.e. the closed Markovian symmetric forms) defined above are strictly quasi- regular (cf., [A,R 98, 90, 91] and section IV-3 of [M,R 92], as well as [A 2003] for the meaning of ”strict quasi regular” ). 2 Denote ( (α), ( (α))) the Dirichlet form on L (S; µ), with the domain ( (α))) defined through TheoremE D 1,E obtained as the closed extension of the closable MarkovianD E symmetric ∞ form (α), understood as first defined on C0 . We shall use the same notation (α) for the closableE form and the closed form. F E For each i N, we denote by X the random variable (i.e., measurable function) on ∈ i (S, (S),µ) , that represents the coordinate x of x =(x , x ,... ), precisely, B i 1 2 X : S x x R. (1.9) i ∋ 7−→ i ∈ By making use of the random variable Xi, we have the following probabilistic expression:

I (x ) µ(dx)= µ(X B), for B (S). (1.10) B i i ∈ ∈ B ZS

2 Theorem 2 (The strict quasi-regularity) Let S = l , for 0 <α 1, let ( (α), ( (α))) (βi) ≤ E D E be the closed Markovian symmetric form on L2(S; µ) given by Theorem 1. If there exists a 1 ∞ −1 − 2 2 positive sequence γ N such that γ < (i.e., γ N is a positive l sequence), { i}i∈ i=1 i ∞ { i }i∈ and an 0 < M0 < , and both ∞ P ∞ 1+α 1 − 1 (β γ ) 2 µ β 2 X > M γ 2 < , (1.11) i i · i | i| 0 · i ∞ i=1 X − 1 − 1 µ X M β 2 γ 2 , i N =1, (1.12) | i| ≤ · i i ∀ ∈ M∈N [ hold, then ( , ( )) is a strictly quasi-regular Dirichlet form. E(α) D E(α) 5 Next, from [A,Kagawa,Yahagi,Y 2020], we quote a theorem corresponding to the Markov processes associated to the non-local Dirichlet forms defined above. Let ( (α), ( (α))), 0 < α 1, be the family of strictly quasi-regular Dirichlet forms on L2(S; µ)E withD theE state space ≤S defined by Theorems 1 and 2. For the strictly quasi-regular Dirichlet form ( , ( )) there exists a properly asso- E(α) D E(α) ciated S-valued Hunt process (see Definitions IV-1.5, 1.8 and 1.13, Theorem V-2.13 and Proposition V-2.15 of [M,R 92])

M Ω, , (X ) , (Px)x . (1.13) ≡ F t t≥0 ∈S△ is a point adjoined to S as an isolated point of S△ S . Let (Tt)t≥0 be the strongly△ continuous contraction semigroup associated with≡ ( ∪ {△}, ( )), and (p ) be E(α) D E(α) t t≥0 the corresponding transition semigroup of kernels of the Hunt process (Xt)t≥0, then for any u C∞ ( ) the following holds: ∈F 0 ⊂D E(α) d d p u (x) µ(dx)= T u, 1) 2 = (T u, 1)=0. (1.14) dt t dt t L (S;µ) E(α) t ZS By this, we see that

p u (x) µ(dx)= u(x) µ(dx), t 0, u C∞, (1.15) t ∀ ≥ ∀ ∈F 0 ZS ZS and hence, by the density of C∞ in L2(S; µ) F 0

Px(X B) µ(dx)= µ(B), B (S), t 0. (1.16) t ∈ ∀ ∈ B ∀ ≥ ZS Thus, the following Theorem 3 holds. Theorem 3 (Associated Markov process) Let 0 <α 1, and let ( , ( )) be a ≤ E(α) D E(α) strictly quasi-regular Dirichlet form on L2(S; µ) that is defined through Theorem 2. Then to ( , ( )), there exists a properly associated S-valued Hunt process M defined by (1.13), E(α) D E(α) the invariant measure of which is µ (cf. (1.16)). As the final preparations for the main discussions given in the next section, we recall the formulation corresponding to the stochastic quantization of the Euclidean free quantum field, discussed in the section 5 of [A,Kagawa,Yahagi,Y 2020]. Let us recall the Bochner-Minlos theorem stated in a general framework. Let E be a nuclear space ( cf., e.g., Chapters 47-51 of [Trèves 67]). Suppose in particular that E is a countably Hilbert space, characterized by a sequence of real Hilbert norms , n N 0 k kn ∈ ∪{ } such that 0 < 1 < < n < . Let En be the completion of E with respect to the norm k k , thenk k by definition··· k kE = ··· E and E E E . Define k kn n≥0 n 0 ⊃ 1 ⊃···⊃ n ⊃··· E∗ the dual space of E ,T and assume the identification E∗ = E . n ≡ n 0 0 6 Then we have

E E E E = E∗ E∗ E∗ E∗. ⊂···⊂ n+1 ⊂ n ⊂···⊂ 0 0 ⊂···⊂ n ⊂ n+1 ⊂···⊂ Since by assumption E is a nuclear space, for any m N 0 there exists an n N 0 , n > m, such that the (canonical) injection T n : E ∈ E∪{ is} a trace class (nuclear∈ ∪{ class)} m n → m positive operator. The Bochner-Minlos theorem (cf. [Hida 80]) is given as follows:

Theorem 4 (Bochner-Minlos Theorem) Let C(ϕ), ϕ E, be a complex valued function on E such that ∈ i) C(ϕ) is continuous with respect to the norm m for some m N 0 ; ii) (positive deﬁniteness) forany k N, k·k ∈ ∪{ } ∈ k α¯ α C(ϕ ϕ ) 0, α C, ϕ E, i =1,...,k; i j i − j ≥ ∀ i ∈ ∀ i ∈ i,j=1 X (where α¯ means complex conjugate of α). iii) (normalization) C(0) = 1. Then, there exists a unique Borel probability measure ν on E∗ such that

C(ϕ)= ei<φ,ϕ>ν(dφ), ϕ E. ∗ ∈ ZE n Moreover, if the (canonical) injection Tm : En Em, for all n > m, is a Hilbert-Schmidt ∗ → ∗ operator, then the support of ν is in En, where < φ,ϕ >= E < φ,ϕ >E is the dualization between φ E∗ and ϕ E. ✷ ∈ ∈

Remark 1.2 The assumption on the continuity of C(ϕ) on E given in i) of the above Theorem 4 can be replaced by the continuity of C(ϕ) at the origin in E, which is equivalent to i) under the assumption that C(ϕ) satisﬁes ii) and iii) in Theorem 4 (cf. e.g., [ItˆoK. 76]). Namely, under the assumption of ii) and iii), the following is equivalent to i): For any ǫ> 0 there exists a δ > 0 such that

C(ϕ) 1 < ǫ, ϕ E with ϕ < δ. | − | ∀ ∈ k km

This can be seen as follows: Assume that ii) and iii) hold. For ii), let k = 3, α1 = α, α2 = α, α3 = β, ϕ1 = 0, ϕ2 = ϕ and ϕ3 = ψ, then by the assumption ii), the positive deﬁniteness− of C, we have

αα (2C(0) C(ϕ) C( ϕ)) · − − − +αβ (C( ψ ϕ) C( ψ)) + αβ (C(ψ + ϕ) C(ψ)) + ββ C(0) 0. · − − − − · − · ≥

7 By making use of the fact that C( ϕ) = C(ϕ), which follows from ii), and the assumption iii), from the above inequality we have−

2 C(ϕ) C(ϕ) C(ψ + ϕ) C(ψ) 0 det − − − . ≤ C(ψ + ϕ) C(ψ) 1 − From this it follows that

C(ψ + ϕ) C(ψ) 2 2 C(ϕ) 1 . | − | ≤ | − |

By making use of the support property of ν by means of the Hilbert-Schmidt operators given by Theorem 4, we can present a framework by which Theorems 1, 2, 3 and 4 can be applied to the stochastic quantization of Euclidean quantum ﬁelds. d d Now, we deﬁne an adequate countably Hilbert nuclear space 0 (R R) (R ), for a given d N. Let H ⊃ S → ≡ S ∈ 1 f : f = (f, f) 2 < , f : Rd R, measurable (Rd), (1.17) H0 ≡ k kH0 H0 ∞ → ⊃ S n o where

d (f,g)H0 (f,g)L2(R ) = f(x)g(x) dx. (1.18) ≡ d ZR Let us consider the following pseudo diﬀerential operators on (Rd R) (Rd) S → ≡ S 2 d+1 d+1 2 d+1 H ( x + 1) 2 ( ∆+1) 2 ( x + 1) 2 , (1.19) ≡ | | − | | −1 2 − d+1 − d+1 2 − d+1 H ( x + 1) 2 ( ∆+1) 2 ( x + 1) 2 , (1.20) ≡ | | − | | with ∆ the d-dimensional Laplace operator. For each n N, deﬁne ∈ the completion of (Rd) with respect to the norm f , f (Rd), (1.21) Hn ≡ S k kn ∈ S 2 where f n = (f, f)n ( in the case where n = 1, to denote the 1 norm we use the exact k k H 1 1 notation H1 , in order to avoid a confusion between the notation of some L or l norms) with the correspondingk k scalar product

(f,g) =(Hnf,Hng) , f,g (Rd). (1.22) n H0 ∈ S Moreover we deﬁne, for n N: ∈ the completion of (Rd) with respect to the norm f , f (Rd), (1.23) H−n ≡ S k k−n ∈ S where f 2 =(f, f) , with k k−n −n (f,g) = ((H−1)nf, (H−1)ng) , f,g (Rd). (1.24) −n H0 ∈ S 8 Then obviously, for f (Rd), ∈ S f f , f f , (1.25) k kn ≤k kn+1 k k−n−1 ≤k k−n and by taking the inductive limit and setting = , we have the following inclusions: H n∈N Hn T ∗. (1.26) H⊂···⊂Hn+1 ⊂ Hn ⊂···⊂H0 ⊂···⊂H−n ⊂ H−n−1 ⊂···⊂H

The (topological) dual space of n is −n, n N. By the operator H−1 given byH (1.20)H on (R∈d) we can define, on each , n N, the S Hn ∈ bounded symmetric (hence self-adjoint) operators (H−1)k, k N 0 (1.27) ∈ ∪{ } d (we use the same notations for the operators on (R ) and on n). Hence, for the canonical injection S H T n+k : , k,n N 0 , (1.28) n Hn+k −→ Hn ∈ ∪{ } it holds that T n+kf = (H−1)kf , f , k n kn k kH0 ∀ ∈ Hn+k where by a simple calculation by means of the Fourier transform, and by Young’s inequality, −1 −1 2 we see that for each n N 0 , H on n is a Hilbert-Schmidt operator and hence (H ) on is a trace class∈ operator.∪{ } H Hn Now, by applying to the strictly positive self-adjoint Hilbert-Schmidt (hence compact) −1 2 d operator H , on 0 = L (R R) the Hilbert-Schmidt theorem (cf., e.g., Theorem VI 16, Theorem VI 22 ofH [Reed,Simon→ 80]) we have that there exists an orthonormal base (O.N.B.) ϕi i∈N of 0 such that { } H H−1ϕ = λ ϕ , i N, (1.29) i i i ∈ where λ N are the corresponding eigenvalues such that { i}i∈ 2 2 0 < <λ <λ 1, which satisfy (λ ) < , i.e., λ N l , (1.30) ··· 2 1 ≤ i ∞ { i}i∈ ∈ i∈N X and ϕi i∈N is indexed adequately corresponding to the finite multiplicity of each λi, i N. By the{ definition} (1.21), (1.22), (1.23) and (1.24) (cf. also (1.27)), for each n N 0 ∈, ∈ ∪{ } n (λ ) ϕ N is an O.N.B. of (1.31) { i i}i∈ Hn and −n (λ ) ϕ N is an O.N.B. of (1.32) { i i}i∈ H−n Thus, by denoting Z the set of integers, by the Fourier series expansion of functions in m, m Z (cf. (1.21)-(1.24)), such that for f , we have H ∈ ∈ Hm f = a (λmϕ ), with a f, (λmϕ ) , i N. (1.33) i i i i ≡ i i m ∈ i∈N X 9 In particular for f (Rd) , it holds that a = λ−m(f, ϕ ) . Moreover we have ∈ S ⊂ Hm i i i H0 a2 = f 2 , i k km i N X∈ that yields an isometric isomorphism τ for each m Z such that m ∈ m m 2 − τm : m f (λ1 a1,λ2 a2,... ) l λ 2m , (1.34) H ∋ 7−→ ∈ ( i ) 2 2 −2m where l −2m is the weighted l space defined by (1.1) with p = 2, and βi = λi . Precisely, (λi ) for f = a (λmϕ ) and g = b (λmϕ ) , with a f, (λmϕ ) , i∈N i i i ∈ Hm i∈N i i i ∈ Hm i ≡ i i m b g, (λmϕ ) , i N, by τ the following holds (cf. (1.31) and (1.32)): i ≡ Pi i m ∈ m P −m m −m m (f, g)m = ai bi = λ (λ ai) λ (λ bi)= τmf, τmg 2 . i i i i l − · · λ 2m i∈N i∈N ( i ) X X By the map τm we can identify, in particular, the two systems of Hilbert spaces given by (1.35) and (1.36) through the following diagram:

... = L2(Rd) ... (1.35) H2 ⊂ H1 ⊂ H0 ⊂ H−1 ⊂ H−2

kk k kk 2 2 2 2 2 − − ... l λ 4 l λ 2 l l(λ2) l(λ4) .... (1.36) ( i ) ⊂ ( i ) ⊂ ⊂ i ⊂ i Example 0. (The Euclidean free quantum field) This fundamental example, which has been considered in [A,Kagawa,Yahagi,Y 2020]), shows how the abstract Theorems 1, 2 and 3, from which we can construct weighted l2-space valued non-local symmetric Markov processes through the non-local Dirichlet forms, can be applied to construct separable Hilbert space (cf. (1.35) and (1.36)) valued Markov processes, which is a stochastic quantization of a given physical random field. Let ν be the Euclidean free field probability measure on ′ ′(Rd). It is characterized 0 S ≡ S by the (generalized) characteristic function C(ϕ) in Theorem 4 of ν0 given by

1 2 −1 d C(ϕ) = exp( (ϕ, ( ∆+ m ) ϕ) 2 d ). for ϕ (R R), (1.37) −2 − 0 L (R ) ∈ S → ′ Equivalently, ν0 is the centered Gaussian probability measure on , the covariance of which is given by S

2 −1 Rd R < φ,ϕ1 > < φ,ϕ1 > ν0(dφ)= ϕ1, ( ∆+ m0) ϕ2 L2 Rd , ϕ1, ϕ2 ( ), ′ · − ( ) ∈ S → ZS (1.38) where ∆ is the d-dimensional Laplace operator and m > 0 ( for d 3, we can also allow 0 ≥ for m = 0) is a given mass for this scalar field. φ(f) =< φ,f >, f (Rd R) is the 0 ∈ S → 10 coordinate process φ to ν0 (for the Euclidean free field cf. [Pitt 71], [Nelson 74], and, e.g., [Simon 74], [Glimm,Jaffe 87], [A,Y 2002], [A,Ferrario,Y 2004]). By (1.37), the functional 2 d C(ϕ) is continuous with respect to the norm of the space 0 = L (R ), and the kernel of 2 −1 H2 2 −1 d ( ∆+ m0) , which is the Fourier inverse transform of ( ξ + m0) , ξ R , is explicitly given− by Bessel functions (cf., e.g., section 2-5 of [Mizohata| | 73]). Then, by∈ Theorem 4 and (1.28) the support of ν0 can be taken to be in the separable Hilbert spaces −n, n 1 (cf. (1.35) and (1.36)). H ≥ 1 Let us apply Theorems 1, 2 and 3 with p = 2 to this random field. We start the consideration from the case where α = 1, a simplest situation. Then, we shall state the corresponding results for the cases where 0 <α< 1. Now, we take ν0 as a Borel probability measure on −2. By (1.34), (1.35) and (1.36), by taking m = 2, τ defines an isometric isomorphism suchH that − −2 2 −2 τ−2 : −2 f (a1, a2,... ) l λ4 , with ai (f, λi ϕi)−2, i N. (1.39) H ∋ 7−→ ∈ ( i ) ≡ ∈ 2 Define a probability measure µ on l 4 such that (λi ) −1 2 µ(B) ν0 τ−2 (B) for B (l λ4 ). ≡ ◦ ∈ B ( i ) 2 4 We set S = l 4 in Theorems 1, 2 and 3, then it follows that the weight βi satisfies βi = λi . (λi ) − 1 We can take γi 2 = λi in Theorem 2, then, from (1.30) we have ∞ ∞ ∞ 1 − 1 β γ µ β 2 X > M γ 2 β γ = (λ )2 < (1.40) i i · i | i| · i ≤ i i i ∞ i=1 i=1 i=1 X X X (1.40) shows that the condition (1.11) holds. Also, as has been mentioned above, since ν ( ) = 1, for any n 1, we have 0 H−n ≥ 1 1 1 2 − 2 − 2 4 − 2 1= ν0( −1)= µ(l λ2 )= µ Xi Mβi γi , i N , for βi = λi , γi = λi. H ( i ) {| | ≤ ∀ ∈ } M∈N [ This shows that the condition (1.12) is satisfied. 2 Thus, by Theorem 2 and Theorem 3, for α = 1, there exists an l 4 -valued Hunt process ((λi) )

M Ω, , (X ) , (Px)x , (1.41) ≡ F t t≥0 ∈S△ associated to the non-local Dirichlet form ( (α), ( (α))). We can now define an -valued processE (YD) E such that H−2 t t≥0 (Y ) τ −1(X ) . (1.42) t t≥0 ≡ −2 t t≥0 2 x Equivalently, by (1.39) for Xt = (X1(t),X2(t),... ) l λ4 , P a.e., by setting Ai(t) such ∈ ( i ) − that A (t) λ X (t) (cf. (1.33) and (1.34)), we see that Y is also given by i ≡ i i t −2 Y = A (t)(λ ϕ )= X (t)ϕ , t 0, Px a.e., for any x S . (1.43) t i i i i i ∈ H−2 ∀ ≥ − ∈ △ i∈N i∈N X X 11 By (1.16) and (1.39), Yt is an −2-valued Hunt process that is a stochastic quantization (according to the definition weH gave to this term) with respect to the non-local Dirichlet form ( ˜ , ( ˜ )) on L2( , ν ), that is defined through ( , ( )), by making use of E(α) D E(α) H−2 0 E(α) D E(α) τ−2. This holds for α = 1. For the cases where 0 <α< 1, we can also apply Theorems 1, 2 and 3, and then have the corresponding result to (1.43). For this purpose we have only to notice that for α (0, 1) if 2 6 ∈ we take ν0 as a Borel probabilty measure on −3, and set S l λ6 , βi λi , γi λi, and H ≡ ( i ) ≡ ≡ define −3 −3 2 τ−3 : −3 f (λ1 a1,λ1 a2,... ) l λ6 , (1.44) H ∋ 7−→ ∈ ( i ) with a (f, λ−3ϕ ) , i N, i ≡ i i −3 ∈ (cf. (1.34), (1.35), (1.36) and (1.39)), then

∞ ∞ α+1 2(α+1) (β γ ) 2 = (λ ) < . i i i ∞ i=1 i=1 X X As a consequence, for α (0, 1), we then see that by the this setting (1.11) and (1.12) also ∈ hold (cf. (1.40) together with the formula given below (1.40)). 2 Deﬁne a probability measure µ on l 6 such that (λi )

−1 2 µ(B) ν τ−3 (B) for B (l λ6 ). (1.45) ≡ ◦ ∈ B ( i ) Then we have an analogue of (1.43) as follows: By Theorem 2 and Theorem 3, for each 2 0 <α< 1, there exists an l 6 -valued Hunt process (λi )

M Ω, , (X ) , (Px)x , (1.46) ≡ F t t≥0 ∈S△ associated to the non-local Dirichlet form ( (α), ( (α))). By making use of M we can deﬁne E D E −1 an −3-valued process (Yt)t≥0 such that (Yt)t≥0 τ−2 (Xt) t≥0, explicitly, by (1.44) for H 2 ≡ −3 x Xt =(X1(t),X2(t),... ) l λ6 , P a.e. x S△, by setting Ai(t) such that Xi(t)= λi Ai(t) ∈ ( i ) − ∈ (cf. (1.33) and (1.34)), then Yt is given by

−3 Y = A (t)(λ ϕ )= X (t)ϕ , t 0, Px a.e., x S . (1.47) t i i i i i ∈ H−3 ∀ ≥ − ∈ △ i∈N i∈N X X By (1.16) and (1.44), Y is an -valued Hunt process that is a stochastic quantization with t H−3 respect to the non-local Dirichlet form ( ˜ , ( ˜ )) on L2( , ν), that is deﬁned through E(α) D E(α) H−3 ( (α), ( (α))) (by making use of τ−3 via (1.45)). E TheD diﬀusionE case α = 2 was already discussed in [A,R 89], [A,R 91] (and references therein).

12 2 Other applications; quantum ﬁeld models with interactions, inﬁnite particle systems

Following analogous arguments to the one used for Example 0 (see the previous section) in the present section we shall treat the following problems, related and complementary to those of [A,Kagawa,Yahagi,Y 2020]: 1. Non-local type stochastic quantization of the (truncated) Høegh-Krohn exponential model with d = 2 (for the considerations on this random field, cf., e.g., [H-K 71], [A,H-K 73], [Simon 74], [Fröhlich,Park 77], [A,Y 2002], [A,Liang,Zegarlinski 2006], [Kusuoka-shi 92], [A,Kawa,Mih,R 2020], [A,DeV,Gu1,2], [A,Hida,Po,R,Str 89]). 2. Non-local type stochastic quantization of the (space cut-off) P (φ)2 and the Albeverio Høegh-Krohn trigonometric model with d = 2 (for the considerations on this random field, cf., e.g., [A.H-K 73], [A,H-K 79], [Brydges,Fröhlich,Sokal 83], [A,H-K,Zegarlinski 89], [A,Y 2002]). 3. Non-local type stochastic quantization of the random fields of classical infinite particle systems (for the considerations on this random field, and its local type stochastic quantizations by means of local Dirichlet form arguments, cf., e.g., [Ruelle 70], [Osada 96], [Tanemura] 97, [Y 96], [A,Kondratiev,Röckner 98]) .

Example 1. (The space cut-off Høegh-Krohn exponential model with d = 2.) Let ν be the Euclidean free field measure on ′ ′(R2), discussed in Example 0 with 0 S ≡ S m0 > 0 (precisely, see (1.37) and (1.38) with d = 2). Here, for simplicity we set the mass 2 term m0 = 1. Let a0 be a given real number and g a given positive valued function on R such that a < √4π, g L2(R2 R) L1(R2 R). (2.1) | 0| ∈ → ∩ → a0 is called ”charge” parameter, g (Euclidean) space cut-off. Define a measurable function V ( ) on the measure space ( ′, ( ′), ν ), as follows: · S B S 0 ∞ (a )n V (φ) V (φ) 0 < g, : φn :>, (2.2) ≡ a0.g ≡ n! n=0 X

(often written as : exp(a0 ) : ), where < , > denotes the dualization between a test function and a distribution, and : φn : denotes the·n·-th Wick monomial of φ, the ′(R2 R)- S → valued random variable of which probability distribution is the free ﬁeld measure ν0 (cf. e.g., [Simon 74]). Then, it is shown (cf. e.g., [A,H-K 74], [Simon74], [A,Y 2002], the ﬁrst work in this direction being for the ”time zero” version [H-K 74]) that

V (φ) Lp( ′; ν ), V (φ) 0, ν a.e.. (2.3) ∈ S 0 ≥ 0 − p≥1 \

13 By (2.3), since 0 e−V (φ) 1, ν a.e., (2.4) ≤ ≤ 0 − it is possible to deﬁne a probability measure ν on ′ such that exp S 1 ν (dφ) e−V (φ) ν (dφ), (2.5) exp ≡ Z 0 where Z is the normalizing constant such that

−V (φ) Z Za0,g e ν0(dφ). (2.6) ≡ ≡ ′ ZS Now, we shall look at the support property of the measure νexp through the Bochner- Minlos theorem (see Theorem 4 in the previous section), and then apply Theorems 1, 2 and 3 in the previous section quoted from [A,Kagawa,Yahagi,Y 2020], to the random field ( ′, ( ′), ν ). To this end, we consider the characteristic function C(ϕ) of ν : S B S exp exp i<φ,ϕ> 2 C(ϕ) e νexp(dφ), ϕ (R R). (2.7) ≡ ′ ∈ S → ZS The existence (i.e., the well definedness) of C(ϕ) and its continuity property can be guaran- teed and shown as follows: We first have the following evaluation:

∞ (i<φ,ϕ>)k νexp(dφ) 1 | ′ k! − | S k Z X=0 ∞ k 1 (i<φ,ϕ>) −V (φ) = e ν0(dφ) |Z ′ k! | S k Z X=1 ∞ k 1 i k −V (φ) <φ,ϕ> e ν0(dφ) (2.8) ≤|Z k! ′ | k S X=1 Z ∞ k 1 i k −V (φ) | | <φ,ϕ> e ν0(dφ) ≤ Z k! ′ | | k=1 ZS X∞ 1 1 k <φ,ϕ> ν0(dφ) ≤ Z k! ′ | | k S X=1 Z 1 ∞ 1 ∞ 1 E [<φ,ϕ>2l]+ E [ <φ,ϕ> l <φ,ϕ> l−1] , ≤ Z { (2l)! ν0 (2l 1)! ν0 | | ·| | } l l X=1 X=1 − where, (2.5) is used for the ﬁrst equality, and Fubini Theorem and (2.4) are applied for the ﬁrst inequality, and (2.4) is again used for the third inequality, and E [ ] denotes the ν0 · expectation with respect to the probability measure ν0. Then, by denoting

2 −1 ϕ (( ∆+1) ϕ,ϕ) 2 R2 k| k| ≡ − L ( ) 14 (from the definition of the Euclidean free field, cf., (1.38)) for the first term of the right hand side of the last inequality of (2.8), it holds that

E [<φ,ϕ>2l]=(2l 1)!! ϕ 2l, ν0 − k| k| and for the second term (with l 2) of the right hand side of the last inequality of (2.8), by the Cauchy Schwarz inequality,≥ it holds that

E [ <φ,ϕ> l <φ,ϕ> l−1] ν0 | | ·| | 2l 2l−2 1 (E [<φ,ϕ> ] E [<φ,ϕ> ]) 2 ≤ ν0 · ν0 2l 2l−2 1 = ((2l 1)!! ϕ (2l 3)!! ϕ ) 2 , − k| k| · − k| k| also, since,

−l 1 −l+1 (2l 1)!! 2 ((2l 1)!! (2l 3)!!) 2 2 − = and − · − , (2l)! l! (2l 1)! ≤ (l 1)! − − we then see the the right hand side of (2.8) is dominated by

∞ −l ∞ −l+1 1 2 2l 2 2l−1 1 1 k|ϕk|2 ϕ + E [ <φ,ϕ> ]+ ϕ (2(e 2 1) + ϕ ). Z { l! k| k| ν0 | | (l 1)!k| k| } ≤ Z − k| k| l=1 l=2 − X X (2.9) (2.8) with (2.9) shows that C(ϕ) is continuous at the origin with respect to the norm

2 −1 2 2 ϕ (( ∆+1) ϕ,ϕ) 2 R2 ϕ 2 2 , ϕ (R R), k| k| ≡ − L ( ) ≤k kL (R ) ∈ S → and hence, by Remark 1.2, C(ϕ) has the same continuity as the characteristic function of the Euclidean free field measure given by (1.37) and (1.38) (though, by the arguments of the present evaluation, in (2.9), the sign of the exponent is positive). Then, by Theorem 4 (with Remark 1.2) and (1.28) the support of νexp can be taken to be in the Hilbert spaces −n, n 1 (cf. (1.35) and (1.36)). H Thus,≥ by repeating the same arguments for the Euclidean free field (cf. (1.39)-(1.43)), and by applying Theorems 1, 2 and 3 in section 1, we have analogous results on the non-local stochastic quantization for the space cut-off Høegh-Krohn field νexp with d = 2 to those for the Euclidean free field with d = 2.

Example 2. (The space cut-off P (φ)2 and the Albeverio Høegh-Krohn trigonometric model with d = 2.) Let us consider once more the Euclidean free field measure ν on ′ ′(R2 R), discussed in Example 0 (precisely, see (1.37) and (1.38) with d = 2). 0 S ≡ S → As in Example 1, for simplicity we set the mass term m0 = 1. Let ν , ν and ν be the probability measures on ′ that are defined by (cf. (2.5)) P (Φ) sin cos S 1 −VP (Φ)(φ) νP (Φ)(dφ) e ν0(dφ), (2.10) ≡ ZP (Φ)

15 and 1 1 −Vsin(φ) −Vcos(φ) νsin(dφ) e ν0(dφ), νcos(dφ) e ν0(dφ), (2.11) ≡ Zsin ≡ Zcos with V (φ) λ < g, : φ2n :>, P (Φ) ≡ for some given λ 0, n N, ≥ ∈ g L2(R2 R) L1(R2 R) satisfying g 0, (2.12) ∈ → ∩ → ≥ and

∞ ( 1)k(a )2k+1 ∞ ( 1)k(a )2k V (φ) λ − 0 < g, : φ2k+1 :>, V (φ) λ − 0 < g, : φ2k :>, sin ≡ (2k + 1)! cos ≡ (2k)! k k X=0 X=0 for some given λ 0, a < √4π, g as in (2.12), (2.13) ≥ | 0| respectively, where ZP (Φ), Zsin and Zcos are the corresponding normalizing constants such that

−VP (φ) −Vsin −Vcos ZP (Φ) e ν0(dφ), Zsin e ν0(dφ), Zcos e ν0(dφ), (2.14) ≡ ′ ≡ ′ ≡ ′ ZS ZS ZS respectively. Under the above settings, it is known that (for the polynomial potential case cf., e.g., [Ne 73a], [Simon 74], [Frö74] [Glimm,Jaffe 84], and for the trigonometric potential case cf. [A.H-K 73], [Frö,Sei 76], [Fröhlich,Park 77], [A,H-K 79], [Glimm,Jaffe 84], [A,Y 2002]), one has that W Lr( ′; ν ) for any ∈ S 0 r≥1 \

−VP (Φ) −Vsin −Vcos W = VP (Φ), Vsin, Vcos, e , e , e . (2.15)

−V The bound (2.4), i.e., 0 e 1, which holds for Vexp in Example 1, does not hold ′ ≤ ≤ for these V s in the present example, but by (2.15), similar to the case of Vexp, we can show that the characteristic functions (cf. Theorem 4) of these probability measures, νP (Φ), νsin and νcos satisfy the comparable continuity as the characteristic function of the Euclidean free field measure. Thus by Theorem 4 and (1.28) the support of νP (Φ), νsin and νcos can be taken to be in the Hilbert spaces , n 1 (cf. (1.35) and (1.36)). Then, for the present random H−n ≥ fields, we can repeat the arguments for the Euclidean free field (cf. (1.39)-(1.41)), getting corresponding results as the the ones of Example 1. The corresponding continuity of the characteristic functions of the probability measures νP (Φ), νsin and νcos can be seen as follows. Denote by F (φ), φ ′(R2 R), one of the positive random variables e−VP (Φ) /Z , ∈ S → P (φ) e−Vsin /Z and e−Vcos /Z on the probability space ( ′, ( ′), ν ). Then by (2.15), by sin cos S B S 0

16 4 applying the Hölder’s inequality (twice) (with p = 3 , q = 4), we have (cf. (2.8)), for ϕ (R2 R), ∈S≡S → ∞ (i<φ,ϕ>)k F (φ) ν0(dφ) 1 | ′ k! − | ZS k=0 X∞ 1 k <φ,ϕ> F (φ) ν0(dφ) ≤ k! ′ | | k=1 ZS X∞ 4 3 1 k 4 4 4 <φ,ϕ> 3 ν0(dφ) (F (φ)) ν0(dφ) ≤ ′ | | ′ k=1 ZS ZS X∞ 1 1 4k 4 4 4 <φ,ϕ> ν0(dφ) (F (φ)) ν0(dφ) . (2.16) ≤ ′ | | ′ k=1 ZS ZS X Also, by denoting r ϕ (( ∆+1)−1ϕ,ϕ), note that the following evaluations hold: ≡ k| |k ≡ − 1 4k 1 ( (<φ,ϕ>) ν0(dφ)) 4 k! ′ ZS 1 2k 1 1 k 1 = ((4k 1)!! r ) 4 r 2 ((4k)!!) 4 k! − ≤ k! 1 k 1 1 k 2 2 4 2 r ((2k)!!) = 1 (4r) , (2.17) ≤ k! (k!) 2 and − 1 − k 1 (k!) 2 2 2 , for k =2l, l N, (2.18) ≤ (l 1)! ∈ − − 1 − k 1 (k!) 2 2 2 , for k =2l 1, l N, l 2. (2.19) ≤ (l 2)! − ∈ ≥ − Then, by applying (2.17), (2.18) and (2.19) to the right hand side of (2.16) we see that ∞ (i<φ,ϕ>)k F (φ) ν0(dφ) 1 | ′ k! − | ZS k=0 X∞ 1 k 1 2 4 4 1 (4r) (F (φ)) ν0(dφ) ≤ (k!) 2 · S′ k=1 Z X ∞ ∞ 1 1 1 1 1 −l l 1 −l+ l− 4 4 (4r) 2 + 2 (4r) + 2 2 (4r) 2 (F (φ)) ν0(dφ) ≤ (l 1)! (l 2)! · ′ l=1 − l=2 − ZS X X 1 1 1 1 1 2r 4 4 =2r 2 1+ e r 2 1+2 2 r 2 (F (φ)) ν0(dφ) . (2.20) · ′ ZS (2.14) and (2.15), (2.16) with (2.20) show that the characteristic functions of νP (φ), νsin and 1 ν are continuous at the origin with respect to the norm r = ϕ ( ∆+1)−1ϕ,ϕ 2 cos k| |k ≡ − 17 for ϕ ′. Hence, by Remark 1.2, the the characteristic functions (cf. Theorem 4) of ν , ∈ S P (Φ) νsin and νcos satisfy the comparable continuity as the characteristic function of the Euclidean free field measure. Thus we can apply Theorems 1, 2 and 3 to the non-local stochastic quantization of the random fields νP (Φ), νsin and νcos, then for these random fields we also get the analogous statements as the one for the Euclidean free field with d = 2 and the one for the space cut-off Høegh-Krohn model in Example 1 (cf. (1.39)-(1.43)).

Example 3. Non-local stochastic quantization for classical infinite particle systems.) In this example we apply Theorems 1, 2 and 3 to the random fields of classical statistical mechanics considered by [Ruelle 70]. On the local type stochastic quantizations for such random fields, the various considerations have been already made through the arguments of local Dirichlet forms (for the fundamental formulations cf. [Tanemura 97], [Osada 96], [Y 96], [A,Kondratiev,Röckner 98]], and for the corresponding extended considerations cf. [Conache 2018], [Osada 2013] and references therein, also cf. [Lang 77] where a first consideration on the stochastic quantization of such a random field through the arguments of an infinite system of stochastic differential equations is presented). But, as far as we know, there exists no considerations on the non-local type stochastic quantizations for such classical particle systems through the arguments by non-local Dirichlet forms. We first recall the configuration space for the classical infinite particle systems (for an original formulation, cf. [Ruelle 70], and also cf. [Y 96] for their interpretation as a subspace of Radon measures, which will be used in the subsequent discussions of the present example (the notations, e.g., , adopted here are diffent to the ones in [Ruelle 70] and [Y 96])). Define Y Y Y : Rd Z such that Y(y) < for any compact K Rd , Y ≡ | → + y∈K ∞ ⊂ P σ[ ] the σ-field generated by Y Y(y)= m , Y ≡ { | y∈B } B running over the bounded Borel set of Rd, m Z , (2.21) P ∈ + where d N is a given dimension, and Z+ is the set of non-negative integers, Z+ N 0 . On the∈ measurable space ( , σ[ ]), suppose that we are given a probability≡ measure∪{ }ν that satisfies (cf. Corollary 2.8,Y Prop.Y 5.2 of [Ruelle 70], and cf. also (2.19) of [Y 96]):

ν UN =1, (2.22) N∈N [ and for some given γ > 0 and real δ,

∞ l ν U c exp[ (γN 2 eδ)] +1, (2.23) N ≤ − − l=0 X 18 where, for N N, ∈ U Y l Z , n(Y,r)2 N 2(2l + 1)d (2.24) N ≡ ∈Y|∀ ∈ + ≤ r:|r|≤l X with n(Y,r) Y(y), (2.25) ≡ y Q X∈ r 1 1 Q y =(y1, ,yd) Rd rj yj

σ[ ] B B σ[ ] . (2.28) Y0 ≡{ ∩Y0 | ∈ Y }

By (2.22) and (2.23), we restrict ν (originally deﬁned on ) to 0 , and denote the restriction by the same notation ν. Then, we can deﬁne theY probabilityY ⊂ space Y

( , σ[ ], ν). (2.29) Y0 Y0 Subsequently, we shall interpret the probability measure ν on the conﬁguration space Y0 to a probability measure on a subset of the space of Radon measures on Rd. We note that each Y can be identiﬁed with z, an element of positive integer valued Radon measures on Rd, such∈Y that (cf. [Y 96])

∞ z = m δ , y Rd, i N, (2.30) i yi i ∈ ∈ i=1 X d for given Y with yi i∈N y R Y(y) = 0 and mi = Y(yi), where δyi denotes the Dirac measure∈Y at the point{ } y ≡{Rd.∈ Deﬁne| 6 } i ∈ ˜ z z corresponds with an Y by (2.30) , (2.31) Y0 ≡ | ∈Y0 σ[ ˜ ] z z corresponds with an Y B, by (2.30) B σ[ ] , (2.32) Y0 ≡ | ∈ ∈ Y0 n o and

ν˜(B˜) ν(B), for B˜ = z z corresponds with an Y B, by (2.30) σ[ ˜ ]. (2.33) ≡ | ∈ ∈ Y0

19 Then, from (2.29), through (2.30)-(2.33), we can define the probability space (on a subset of the space of Radon measures (cf., e.g., [Kallenberg 83], also as a general reference cf. [Trèves 67] ) such that ( ˜ , σ[ ˜ ], ν˜). (2.34) Y0 Y0 Next, we embed ˜ defined by (2.31) into a Hilbert space, and interpret the present Y0 random field ( ˜0, σ[ ˜0], ν˜) to be the one on which we can apply Theorems 1, 2 and 3. To this end, for theY presentY consideration, we modify the Hilbert-Schmidt operator and the corresponding nuclear space defined through (1.19)-(1.32) as follows: Let

L2(Rd R; λ), with λ the Lebesgue measure on Rd, (2.35) H0 ≡ → and 2 d+1 d+1 2 d+1 H˜ ( x + 1) ( ∆+1) 2 ( x + 1) , (2.36) ≡ | | − | | −1 2 −(d+1) − d+1 2 −(d+1) H˜ ( x + 1) ( ∆+1) 2 ( x + 1) , (2.37) ≡ | | − | | ˜ and ˜ be the completion of (Rd R), the space of Schwartz’s rapidly decreasing Hn H−n S ≡ S → functions, equipped with the norms corresponding to the inner products ( , )n and ( , )−n respectively such that · · · · (f,g) (H˜ nf, H˜ ng) , f,g , (2.38) n ≡ H0 ∈ S (f,g) ((H˜ −1)nf, (H˜ −1)ng) , f,g . (2.39) −n ≡ H0 ∈ S Then, through arguments analogous to those performed in section 1 and in the previous examples (with obvious adequate modiﬁcations of notations and notions) we have the continuous inclusion

˜ ˜ ˜ = L2 ˜ ˜ ˜ , (2.40) H3 ⊂ H2 ⊂ H1 ⊂ H0 ⊂ H−1 ⊂ H−2 ⊂ H−3 −1 and by the self-adjoint extension of H on , for each domain ˜n, n Z (setting ˜0 = 0), we also have the Hilbert-Schmidt operator SH˜ −1. H ∈ H H Let ϕ˜ N be the orthonormal base of the Hilbert space (cf. (1.29)-(1.32)) such that { i}i∈ H0 H˜ −1ϕ˜ = λ˜ ϕ˜ , i N, (2.41) i i i ∈

where λ˜ N is the family of the corresponding eigenvalues, that satisﬁes { i}i∈ 2 0 < < λ˜ < λ˜ 1, λ˜ N l . (2.42) ··· 2 1 ≤ { i}i∈ ∈

Through the preparations (2.35)-(2.42) above, we see that for any Y 0 the corresponding Radon measure z ˜ deﬁned by (2.30) satisﬁes ∈ Y ∈ Y0 z ˜ . (2.43) ∈ H−1 Equivalently, we are able to show that the following Lemma 2.1 holds:

20 Lemma 2.1 For the subset of Radon measures ˜ deﬁned by (2.31), it holds that Y0 ˜ ˜ . Y0 ⊂ H−1 The proof of Lemma 2.1 is given in Section 3 Appendix. Moreover, the following Lemma 2.2 holds.

Lemma 2.2 For the σ-field σ[ ˜0] defined by (2.32) and the Borel field ( ˜−r) of the Hilbert space ˜ , it holds that Y B H H−r σ[ ˜ ] ( ˜ ) ˜ , for r 1. Y0 ⊃ B H−r ∩ Y0 ≥ The proof of Lemma 2.2 is also given in the Appendix. By Lemmas 2.1 and 2.2, the probability measure of the classical infinite particle system ν˜ on ( ˜ , σ[ ˜ ]) (cf. (2.34)) can be extended, for each r 1, to a Borel probability measure Y0 Y0 ≥ ν on ( ˜ , ( ˜ ) as follows: r H−r B H−r ν (B)=˜ν(B ˜ ), B ( ˜ ). (2.44) r ∩ Y0 ∈ B H−r For the subsequent discussion, we take r = 3, and consider the corresponding extended random field ( ˜−3, ( ˜−3), ν3) to ( ˜0, σ[ ˜0], ν˜) (cf. (2.34)). By makingH use ofB LemmaH 2.2, weY shallY proceed to the application of Theorems 1, 2 and 3 to the random field ( ˜−3, ( ˜−3), ν3). From (2.40), (2.41) and (2.42) (cf. (1.29)-(1.34)), we see that for k =0, 1H, 2, 3 withB H ˜ = ˜ = , H−0 H0 H k (λ˜ ) ϕ˜ N is an O.N.B. of ˜ , (2.45) { i i}i∈ Hk −k (λ˜ ) ϕ˜ N is an O.N.B. of ˜ , , (2.46) { i i}i∈ H−k and we can define an isometric isomorphism τ between ˜ and l2(λ˜6) such that H−3 τ : ˜ f (λ˜−3a , λ˜−3a ,... ) l2(λ˜6), with a (f, λ˜−3ϕ˜ ) , i N, (2.47) H−3 ∋ 7−→ 1 1 2 2 ∈ i ≡ i i −3 ∈ where the inner product ( , ) is defined, not by (1.24), but by (2.39). Then, through ν · · −3 p defined by Lemma 2.2 and the mapping τ, we can define a Borel probability measure µ on l2(λ˜6) as follows: µ(B) ν τ −1(B) for B (l2(λ˜6)). (2.48) ≡ 3 ◦ ∈ B Now, by setting S = l2(λ˜6) in Theorems 1, 2 and 3, for each α (0, 1], we have an l2(λ˜6)- ∈ valued Hunt process M (Ω, , (Xt)t≥0, (Px)x∈S△ ), associated to the non-local Dirichlet 2≡ F form ( (α), ( (α))) on L (S,µ). We can then define an ˜−3-valued Hunt process (Yt)t≥0 (cf. (1.42)E andD (1.43))E such that H

i∈N Xi(t)˜ϕi, Xi(t) = ∆ Yt = 6 (2.49) ′ ( P∆ , Xi(t) = ∆, where ∆′ is a point adjoint to ˜ (the cemetery). H−3 21 Remark 2.1 i) These considerations performed in Example 3 is adapted to all the case α (0, 1], but, if we restrict our discussions to the case where α = 1, then we are able ∈ 2 ˜4 2 ˜4 to take S = l (λ ), and have an l (λ )-valued Hunt process M (Ω, , (Xt)t≥0, (Px)x∈S△ ), associated to the non-local Dirichlet form ( , ( )) on L2(S,µ≡ ), andF then we can define E(α) D E(α) an ˜ -valued Hunt process (Y ) (cf. (2.49)) through the same discussion as Example 0. H−2 t t≥0 ii) In order to consider another jump type Markov processes, which are natural analogues of the diffusion process, with invariant measure ν˜ defined by (2.33) and (2.34), constructed through the local type Dirichlet form defined by [Y 96], where the present ν˜ was denoted by µ, we should define, analogous to [Y 96], a corresponding non-local type Dirichlet forms by making use of a system of density distributions (cf. Assumption 1 and Remark 1 of [Y 96]). These would be different from the non-local type Dirichlet forms discussed in the present paper, since they would involve the mentioned system of density distributions.

3 Appendix

Proof of Lemma 2.1. From (2.22) and (2.23), the assumption for the original measure ν, Lemma 2.1 can be proven as follows: It suﬃces to show that (2.43) holds for any z that corresponds with an Y U for some N N (see (2.27)). ∈ N ⊂Y0 ∈ For Y U with N N, let z = ∞ m δ ˜ , (3.1) ∈ N ⊂Y0 ∈ i=1 i yi ∈ Y0 the Radon measure corresponding with Y through (2.30). Then,P for any test function ϕ ∈ S (denoting by the dualization between the distribution z and the test function ϕ),

22 we have

∞ ∞ = m ϕ(y ) = m ϕ(y ) | | | i i | | i i | i=1 l X X=0 yi∈QXr:|r|=l ∞ ∞ n(Y,r)( sup ϕ(y) ) (n2(Y,r))( sup ϕ(y) ) ≤ y∈Q | | ≤ y∈Q | | l r l r X=0 |Xr|=l X=0 |Xr|=l ∞ (n2(Y,r))( sup ( y 2 + 1)d+1ϕ(y) ) ≤ y∈Q | | | | l r X=0 |Xr|=l ∞ 1 2 d+1 2 d+1 2 (n (Y,r)) (( ∆+1) 2 ( y + 1) ϕ(y)) dy 2 ≤ − | | l Qr X=0 |Xr|=l Z ∞ 1 (n2(Y,r)) (( y 2 + 1)−(d+1))2dy 2 ≤ | | l=0 |r|=l ZQr X X 1 2 d+1 d+1 2 d+1 2 ( y + 1) (( ∆+1) 2 ( y + 1) ϕ(y) dy 2 , × Qr | | − | | Z∞ 1 2 2 −(d+1) 2 2 (n (Y,r)) (( y + 1) ) dy ϕ ˜ . (3.2) ≤ | | k kH1 l=0 |r|=l ZQr X X In the above deductions, to get the third inequality we have applied the Sobolev’s embedding m,2 d theorem (cf., e.g., [Mizohata 73]) that gives for the Sobolev space W with m = [ 2 ] + 1, m,2 d d that W Cb(R ) (cf. the explanation given below (3.10)), where Cb(R ) denotes the space of real⊂ valued bounded continuous functions on Rd. Since, for r Zd by denoting r = l, for some C < it holds that ∈ | | ∞

2 −(d+1) 2 2 − 3d+5 ( y + 1) dt C(l + 1) 2 , (3.3) | | ≤ ZQr by this together with (3.1), we can evaluate the right hand side of (3.2). We consequently see that the following holds for some constants C , C ,C < (only C depends on N): 1 2 3 ∞ 3 ∞ 2 −d 2 − 3d+5 + d 2 = C1 n (Y,r)(2l + 1) (l + 1) 4 2 + N ϕ ˜ | | k kH1 l=1 ∞ X 2 − d − 5 2 C2 N (l + 1) 2 2 + N ϕ ˜ C3 ϕ ˜ , ϕ ˜1. (3.4) ≤ k kH1 ≤ k kH1 ∀ ∈ H l=1 X (3.4) shows that z ˜∗ = ˜ , (3.5) ∈ H1 H−1

23 for any z that corresponds with an Y UN 0 for some N N. Since N N is arbitrary, the proof of (2.43) is completed. ∈ ⊂Y ∈ ∈ Proof of Lemma 2.2. Let r 1. We shall show that ≥ σ[ ˜ ] ( ˜ ) ˜ . (3.6) Y0 ⊃ B H−r ∩ Y0 Since ˜−r is a separable Hilbert space, and hence, it is a Souslin space, and also since the dual spaceH of ˜ is ˜ (see (2.40)), it holds that (cf. e.g., [Ba 70]) H−r Hr ( ˜ )= σ[ ˜ ] the σ-field generated by z z ˜ ,ϕ(z) < t , t R, ϕ ˜ , (3.7) B H−r Hr ≡ { | ∈ H−r } ∈ ∈ Hr where ϕ(z) =< z,ϕ > denotes the dualization between the distribution z ˜−r and the test function ϕ ˜ (cf. (3.2)). To see that (3.6) holds, by (3.7) it suffices to∈ showH that ∈ Hr z z ˜ , ϕ(z) < t ˜ σ[ ˜ ], for any ϕ ˜ and t R. (3.8) { | ∈ H−r } ∩ Y0 ∈ Y0 ∈ Hr ∀ ∈ By Lemma 2.1, since ˜ ˜ , and it holds that H−r ⊃ Y0 z z ˜ , ϕ(z) < t ˜ = z z ˜ ˜ , ϕ(z) < t = z z ˜ , ϕ(z) < t , { | ∈ H−r } ∩ Y0 { | ∈ H−r ∩ Y0 } { | ∈ Y0 } we see that (3.8) is equivalent to the following: z z ˜ , ϕ(z) < t σ[ ˜ ], for any ϕ ˜ and t R. (3.9) { | ∈ Y0 } ∈ Y0 ∈ Hr ∀ ∈ On the other hand, by (2.38), from the Sobolev’s embedding theorem (cf., e.g., Th. 3.15 in [Mizohata 73]), since C (Rd R) W r(d+1),2(Rd), it holds that b → ⊂ C˜ ( x 2 + 1)−r(d+1)ψ ψ C (Rd R) ˜ , (3.10) ≡{ | | | ∈ b → } ⊃ Hr r(d+1),2 where Cb denotes the space of real valued bounded continuous functions, and W denotes the Sobolev space defined., e.g., by Def. 2.9 of [Mizohata 73], where the notation such that r(d+1) = W r(d+1),2 is adopted. Thus, by (3.10), in order to prove (3.9), that is EL2 equivalent to (3.8), it suffices to show that z z ˜ , ϕ(z) < t σ[ ˜ ], ϕ C,˜ t R. (3.11) { | ∈ Y0 } ∈ Y0 ∀ ∈ ∀ ∈ For (3.11), we used the fact that C˜ can be taken as the dual space of ˜0, which is included in the proof of Lemma 2.1 (cf. (3.2)), but is easily seen as follows:Y By (3.10), for ϕ = ( x 2 + 1)−r(d+1)ψ C˜ with ψ C and any z Y˜ , it holds that | | ∈ ∈ b ∈ 0 ∞ ∞ ϕ(z) = m ϕ(y ) = m ϕ(y ) | | | i i | | i i | i=1 l X X=0 yi∈XQt:|t|=l ∞ ∞ 2 2 −r(d+1) n(Y, t)(sup ϕ(y) ) ψ L∞ (n (Y, t))(sup ( y + 1) ≤ y∈Q | | ≤k k y∈Q | | | l t l t X=0 X|t|=l X=0 X|t|=l ∞ 2 2 1 −r(d+1) 2 ψ ∞ (n (Y, t))((t ) + n (Y, 0) < , (3.12) ≤ k kL − 2 ∞ l=1 |t|=l X X 24 where the last equality follows from (2.24) and (2.27). In addition, note that for ϕ C˜, by the decomposition such that ϕ = ϕ ϕ , where ∈ + − − ϕ (x) max ϕ(x), 0 and ϕ (x) max ϕ(x), 0 , it holds that + ≡ { } − ≡ {− } ϕ , ϕ C.˜ (3.13) + − ∈ Also, note that the following holds:

z z ˜ , ϕ(z) < t { | ∈ Y0 } = z z ˜ , ϕ (z) ϕ (z) < t { | ∈ Y0 + − − } = f z ˜ , ϕ (z) < t + s z z ˜ , ϕ (z) >s , (3.14) { | ∈ Y0 + }∩{ | ∈ Y0 − } s∈Q [ where Q denotes the ﬁeld of rational numbers. Thus, since the right hand side of (3.14) is a countable operation, from (3.13) and (3.14), to prove (3.11) it suﬃces to show that the following holds:

d z z ˜0, ϕ(z) < t σ[ ˜0], for any ϕ C˜ such that ϕ(x) 0, x R , and t R. { | ∈ Y } ∈ Y ∈ ≥ ∈ ∀ ∈(3.15) To this end for ϕ C˜ with ϕ(x) 0, x Rd, deﬁne ϕ C˜, n N, that satisfy the ∈ ≥ ∀ ∈ n ∈ ∈ following: 0 ϕ (x) ϕ (x) ϕ(x), x Rd, n N, (3.16) ≤ n ≤ n+1 ≤ ∀ ∈ ∀ ∈ supp [ϕ ] x x Rd, x n , (3.17) n ⊂ { | ∈ | | ≤ } lim ϕn ϕ L∞ =0, (3.18) n→∞ k − k then, since z ˜ is a non-negative (integer)-valued Radon measure on Rd (cf. (2.30)), we ∈ Y0 can use an argument of a monotonicity, we have

z z ˜ , ϕ(z) < t = z z ˜ , ϕ (z) < t . (3.19) { | ∈ Y0 } { | ∈ Y0 n } n N [∈ d d For each ϕn C0(R R+), there exists a sequence of simple functions ϕn,k k∈N on (R ), the Borel σ-ﬁeld∈ of R→d, such that { } B

0 ϕ (x) ϕ (x) ϕ (x), x Rd, k N, (3.20) ≤ n,k ≤ n,k+1 ≤ n ∀ ∈ ∈

lim ϕn,k ϕn L∞ =0, (3.21) k→∞ k − k where C (Rd R ) denotes the space of non-negative continuous functions on Rd with 0 → + compact supports. Then, by (3.20), (3.21) and again by the monotonicity of the sequence of sets, that follows from the positivity of z ˜ , it holds that ∈ Y0 z z ˜ , ϕ (z) < t = z z ˜ , ϕ (z) < t . (3.22) { | ∈ Y0 n } { | ∈ Y0 n,k } k N [∈ 25 By the deﬁnition of the σ-ﬁeld σ[ ˜ ], provided through (2.21), (2.28) and (2.32), since Y0 z z ˜ , ϕ (z) < t σ[ ˜ ], n N, k N, { | ∈ Y0 n,k } ∈ Y0 ∀ ∈ ∀ ∈ from (3.22) we have

z z ˜ , ϕ (z) < t σ[ ˜ ], n N, { | ∈ Y0 n } ∈ Y0 ∀ ∈ and thus, from (3.19), we see that (3.15) holds. This complete the proof of (3.6). Acknowledgements The authors would like to gratefully acknowledge the support received from various institutions and grants. In particular for the first and the fifth named author, the conference ”Random transformations and invariance in stochastic dynamics” held in Verona 2019 supported by Dipartimento di Matematica, Universitàdegli Studi di Milano, and its organizers, in particular prof. S. Ugolini, where they could get fruitful discussions with the participants, e.g., prof. F. Guerra, prof. P. Blanchard, prof. D. Elworthy to whom strong acknowledgements are expressed; for the fourth and fifth named authors, IAM and HCM at the University of Bonn, Germany; also for the fourth and fifth named authors, international conference ”mathematical analysis and its application to mathematical physics” held at Samarkand Univ., 2018 supported by Samarkand Univ. Uzbekistan, and its organizer prof. S. N. Lakaev ; for the fifth named author, SFB 1283 and Bielefeld University, Germany; also for the fifth named author, the conference ”Quantum Bio- Informatics” held at Tokyo University of Science 2019 supported by Tokyo University of Science, and its organizer prof. N. Watanabe, where he could get fruitful discussion with prof. L. Accardi to whom a strong acknowledgement is expressed; moreover for the fifth named author, the conference ” Stochastic analysis and its applications” held at Tohoku Univ. supported by the grant 16H03938 of Japan Society for the Promotion of Science, and its organizer prof. S. Aida. Also, the fifth named author expresses his strong acknowledgements to prof. Michael Röckner for several fruitful discussions on the corresponding researches.

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