Abstract
The general framework on the non-local Markovian symmetric forms on weighted lp \((p \in [1, \infty ])\) spaces constructed by Albeverio et al. (Commn. Math. Phys. 388, 659–706, 2021 Kagawa) by restricting the situation where p = 2, is applied to probability measure spaces describing the space cut-off P(ϕ)2 Euclidean quantum field, the 2-dimensional Euclidean quantum fields with exponential and trigonometric potentials, and the measure associated with 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.
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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, Università degli 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.
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Appendix: A
Appendix: A
Proof Proof of Lemma 2.1
From Eqs. 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 Eq. 2.43 holds for any z that corresponds with an \({\mathbb {Y}} \in U_{N} \subset {\mathcal {Y}}_{0}\) for some \(N \in {\mathbb {N}}\) (see Eq. 2.27).
the Radon measure corresponding with \({\mathbb {Y}}\) through Eq. 2.30. Then, for any test function \(\varphi \in {\mathcal {S}}\) (denoting by < z,φ > the dualization between the distribution z and the test function φ), we have
In the above deductions, to get the third inequality we have applied the Sobolev’s embedding theorem (cf., e.g., [56]) that gives for the Sobolev space Wm,2 with \(m=[\frac {d}{2}] +1\), that \(W^{m,2} \subset C_{b}({\mathbb {R}}^{d})\) (cf. the explanation given below (A.10), where \(C_{b}({\mathbb {R}}^{d})\) denotes the space of real valued bounded continuous functions on \({\mathbb {R}}^{d}\). Since, for \(r \in {\mathbb {Z}}^{d}\) by denoting |r| = l, for some \(C < \infty \) it holds that
by this together with Eq. A.1, we can evaluate the right hand side of Eq. A.2. We consequently see that the following holds for some constants C1, C2, \(C_{3} < \infty \) (only C3 depends on N):
Equation A.4 shows that
for any z that corresponds with an \({\mathbb {Y}} \in U_{N} \subset {\mathcal {Y}}_{0}\) for some \(N \in {\mathbb {N}}\). Since \(N \in {\mathbb {N}}\) is arbitrary, the proof of Eq. 2.43 is completed. □
Proof Proof of Lemma 2.2
Let r ≥ 1. We shall show that
Since \({\tilde {{\mathscr{H}}}}_{-r}\) is a separable Hilbert space, and hence, it is a Souslin space, and also since the dual space of \({\tilde {{\mathscr{H}}}}_{-r}\) is \({\tilde {{\mathscr{H}}}}_{r}\) (see Eq. 2.40), it holds that (cf. e.g., [27])
where φ(z) =< z,φ > denotes the dualization between the distribution \(z \in {\tilde {{\mathscr{H}}}}_{-r}\) and the test function \(\varphi \in {\tilde {{\mathscr{H}}}}_{r}\) (cf. Eq. A.2). To see that Eq. A.6 holds, by Eq. A.7 it suffices to show that
By Lemma 2.1, since \({\tilde {{\mathscr{H}}}}_{-r} \supset {\tilde {\mathcal {Y}}}_{0}\), and it holds that
we see that Eq. A.8 is equivalent to the following:
On the other hand, by Eq. 2.38, from the Sobolev’s embedding theorem (cf., e.g., Th. 3.15 in [56]),since \(C_{b}({\mathbb {R}}^{d} \to {\mathbb {R}}) \subset W^{r(d+1),2}({\mathbb {R}}^{d})\), it holds that
where Cb denotes the space of real valued bounded continuous functions, and Wr(d+ 1),2 denotes the Sobolev space defined., e.g., by Def. 2.9 of [56], where the notation such that \({\mathcal {E}}_{L^{2}}^{r(d +1)} = W^{r(d +1),2}\) is adopted. Thus, by Eq. A.10, in order to prove Eq. A.9, that is equivalent to Eq. A.8, it suffices to show that
For Eq. A.11, we used the fact that \({\tilde {C}}\) can be taken as the dual space of \({\tilde {\mathcal {Y}}}_{0}\), which is included in the proof of Lemma 2.1 (cf. Eq. A.2), but is easily seen as follows: By Eq. A.10, for \(\varphi = (|x|^{2} + 1)^{- r(d +1)} \psi \in {\tilde {C}}\) with ψ ∈ Cb and any \(z \in {\tilde {Y}}_{0}\), it holds that
where the last equality follows from Eqs. 2.24 and 2.27.
In addition, note that for \(\varphi \in \tilde {C}\), by the decomposition such that φ = φ+ − φ−, where \(\varphi _{+}(x) \equiv {\max \limits } \{\varphi (x), 0 \}\) and \(\varphi _{-} (x) \equiv {\max \limits } \{ -\varphi (x), 0 \}\), it holds that
Also, note that the following holds:
where \({\mathbb {Q}}\) denotes the field of rational numbers. Thus, since the right hand side of Eq. A.14 is a countable operation, from Eqs. A.13 and A.14, to prove Eq. A.11 it suffices to show that the following holds:
To this end for \(\varphi \in {\tilde {C}}\) with φ(x) ≥ 0, \(\forall x \in {\mathbb {R}}^{d}\), define \(\varphi _{n} \in {\tilde {C}}\), \(n \in {\mathbb {N}}\), that satisfy the following:
then, since \(z \in {\tilde {\mathcal {Y}}}_{0}\) is a non-negative (integer)-valued Radon measure on \({\mathbb {R}}^{d}\) (cf. Eq. 2.30), we can use an argument of a monotonicity, we have
For each \({\varphi }_{n} \in C_{0}({\mathbb {R}}^{d} \to {\mathbb {R}}_{+})\), there exists a sequence of simple functions \(\{{\varphi }_{n,k} \}_{k \in {\mathbb {N}}}\) on \({\mathscr{B}}({\mathbb {R}}^{d})\), the Borel σ-field of \({\mathbb {R}}^{d}\), such that
where \(C_{0}({\mathbb {R}}^{d} \to {\mathbb {R}}_{+})\) denotes the space of non-negative continuous functions on \({\mathbb {R}}^{d}\) with compact supports. Then, by Eqs. A.20, A.21 and again by the monotonicity of the sequence of sets, that follows from the positivity of \(z \in {\tilde {\mathcal {Y}}}_{0}\), it holds that
By the definition of the σ-field \(\sigma [{\tilde {\mathcal {Y}}}_{0}]\), provided through Eqs. 2.21, 2.28 and 2.32, since
from Eq. A.22 we have
and thus, from Eq. A.19, we see that Eq. A.15 holds. This complete the proof of Eq. A.6. □
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Albeverio, S., Kagawa, T., Kawasaki, S. et al. Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces. Potential Anal 59, 1941–1970 (2023). https://doi.org/10.1007/s11118-022-10018-9
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DOI: https://doi.org/10.1007/s11118-022-10018-9
Keywords
- Non local Dirichlet forms on infinite dimensional spaces
- Space cut-off P(ϕ)2
- \(\exp \phi \)
- \({\sin \limits } \phi \)-quantum field models
- Euclidean quantum field
- Infinite particle systems
- Non-local stochastic quantization