The present paper is a continuation of our previous work on the stochastic quantization of the $\... more The present paper is a continuation of our previous work on the stochastic quantization of the $\exp(\Phi)_2$-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full "$L^{1}$-regime" $\vert\alpha\vert<\sqrt{8\pi}$ of the charge parameter $\alpha$. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.
The present paper is a continuation of our previous work on the stochastic quantization of the (Φ... more The present paper is a continuation of our previous work on the stochastic quantization of the (Φ)_2-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full "L^1-regime" |α|<√(8π) of the charge parameter α. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.
The present paper is a continuation of our previous work (Hoshino et al., J Evol Equ 21:339–375, ... more The present paper is a continuation of our previous work (Hoshino et al., J Evol Equ 21:339–375, 2021) on the stochastic quantization of the $$\exp (\Phi )_2$$ exp ( Φ ) 2 -quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full “$$L^{1}$$ L 1 -regime” $$\vert \alpha \vert <\sqrt{8\pi }$$ | α | < 8 π of the charge parameter $$\alpha $$ α . We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.
We consider a quantum field model with exponential interactions on the two-dimensional torus, whi... more We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the (Φ)_2-quantum field model or Høegh-Krohn's model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.
We consider space-time quantum fields with exponential/trigonometric interactions. In the context... more We consider space-time quantum fields with exponential/trigonometric interactions. In the context of Euclidean quantum field theory, the former and the latter are called the Hoegh-Krohn model and the Sine-Gordon model, respectively. The main objective of the present paper is to construct infinite dimensional diffusion processes which solve modified stochastic quantization equations for these quantum fields on the two-dimensional torus by the Dirichlet form approach and to prove strong uniqueness of the corresponding Dirichlet operators.
Dedicated to Professor Kiyosi Ito ̂ on the occasion of his 90th birthday Abstract: In this paper,... more Dedicated to Professor Kiyosi Ito ̂ on the occasion of his 90th birthday Abstract: In this paper, we discuss asymptotics for certain Banach space-valued Ito ̂ functionals of Brownian rough paths based on the results of Inahama-Kawabi [10] and Inahama [9]. Our main tool is the Banach space-valued rough path theory of T. Lyons. As examples, we deal with heat pro-cesses on loop spaces and solutions of the stochastic differential equations (SDEs) on M-type 2 Banach spaces. 1
Abstract. In the present paper, we study an explicit effect of nonsymmetry on asymptotics of the ... more Abstract. In the present paper, we study an explicit effect of nonsymmetry on asymptotics of the n-step transition probability as n → ∞ for a class of non-symmetric random walks on the triangular lattice. Realizing the triangular lattice into R 2 appropriately, we observe that the Euclidean distance in R 2 naturally appears in the asymptotics. We characterize this realization from a geometric view point of Kotani-Sunada's standard realization of crystal lattices. As a corollary of the main theorem, we obtain that the transition semigroup generated by the nonsymmetric random walk approximates the heat semigroup generated by the usual Brownian motion on R 2 .
In this survey paper, we discuss strong uniqueness of Dirichlet operators related to stochastic q... more In this survey paper, we discuss strong uniqueness of Dirichlet operators related to stochastic quantization under exponential (and polynomial) interactions in one-dimensional infinite volume based on joint works with Sergio Albeverio and Michael Rockner (Albeverio et al., J Funct Anal 262:602–638, 2012, [4], Kawabi and Rockner, J Funct Anal 242:486–518, 2007, [11]). We also raise an open problem.
Journal of Mathematical Sciences-the University of Tokyo, 2006
In this paper, we establish the Littlewood-Paley-Stein inequality on general metric spaces under ... more In this paper, we establish the Littlewood-Paley-Stein inequality on general metric spaces under a weaker condition than the lower boundedness ofBakry-Emery's Γ 2. We also discuss Riesz transforms. As examples, we deal with diffusion processes on a path space associated with stochastic partial differential equations (SPDEs in short) and a class ofsuperprocesses with immigration. 1. Framework and Results After the Meyer's celebrated work (16), many authors studied the Lit- tlewood-Paley-Stein inequality by a probabilistic approach. Especially, Shigekawa-Yoshida (20) studied it for symmetric diffusion processes on a general state space. In (20), they assumed the existence ofa suitable core A which is not only a ring but also stable under the operation ofthe semigroup and the infinitesimal generator to employ Bakry-Emery's Γ2-method in the proof, and established the Littlewood-Paley-Stein inequality under that Γ2 is bounded from below. However, it is very difficult to check...
In this article we will give a survey on the Laplace-type asymptotics for the laws of solutions o... more In this article we will give a survey on the Laplace-type asymptotics for the laws of solutions of formal Stratonovich-type stochastic differential equations in Banach spaces. The rigorous meaning of the solutions is given by the rough path theory. The key of the proof is the (stochastic) Taylor expansion, which is, in fact, deterministic in this context. The main example we have in mind is the Brownian motion over loop groups.
We consider a quantum field model with exponential interactions on the twodimensional torus, whic... more We consider a quantum field model with exponential interactions on the twodimensional torus, which is called the exp(Φ)2-quantum field model or HøeghKrohn’s model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.
The present paper is a continuation of our previous work on the stochastic quantization of the $\... more The present paper is a continuation of our previous work on the stochastic quantization of the $\exp(\Phi)_2$-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full "$L^{1}$-regime" $\vert\alpha\vert<\sqrt{8\pi}$ of the charge parameter $\alpha$. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.
The present paper is a continuation of our previous work on the stochastic quantization of the (Φ... more The present paper is a continuation of our previous work on the stochastic quantization of the (Φ)_2-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full "L^1-regime" |α|<√(8π) of the charge parameter α. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.
The present paper is a continuation of our previous work (Hoshino et al., J Evol Equ 21:339–375, ... more The present paper is a continuation of our previous work (Hoshino et al., J Evol Equ 21:339–375, 2021) on the stochastic quantization of the $$\exp (\Phi )_2$$ exp ( Φ ) 2 -quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full “$$L^{1}$$ L 1 -regime” $$\vert \alpha \vert <\sqrt{8\pi }$$ | α | < 8 π of the charge parameter $$\alpha $$ α . We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.
We consider a quantum field model with exponential interactions on the two-dimensional torus, whi... more We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the (Φ)_2-quantum field model or Høegh-Krohn's model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.
We consider space-time quantum fields with exponential/trigonometric interactions. In the context... more We consider space-time quantum fields with exponential/trigonometric interactions. In the context of Euclidean quantum field theory, the former and the latter are called the Hoegh-Krohn model and the Sine-Gordon model, respectively. The main objective of the present paper is to construct infinite dimensional diffusion processes which solve modified stochastic quantization equations for these quantum fields on the two-dimensional torus by the Dirichlet form approach and to prove strong uniqueness of the corresponding Dirichlet operators.
Dedicated to Professor Kiyosi Ito ̂ on the occasion of his 90th birthday Abstract: In this paper,... more Dedicated to Professor Kiyosi Ito ̂ on the occasion of his 90th birthday Abstract: In this paper, we discuss asymptotics for certain Banach space-valued Ito ̂ functionals of Brownian rough paths based on the results of Inahama-Kawabi [10] and Inahama [9]. Our main tool is the Banach space-valued rough path theory of T. Lyons. As examples, we deal with heat pro-cesses on loop spaces and solutions of the stochastic differential equations (SDEs) on M-type 2 Banach spaces. 1
Abstract. In the present paper, we study an explicit effect of nonsymmetry on asymptotics of the ... more Abstract. In the present paper, we study an explicit effect of nonsymmetry on asymptotics of the n-step transition probability as n → ∞ for a class of non-symmetric random walks on the triangular lattice. Realizing the triangular lattice into R 2 appropriately, we observe that the Euclidean distance in R 2 naturally appears in the asymptotics. We characterize this realization from a geometric view point of Kotani-Sunada's standard realization of crystal lattices. As a corollary of the main theorem, we obtain that the transition semigroup generated by the nonsymmetric random walk approximates the heat semigroup generated by the usual Brownian motion on R 2 .
In this survey paper, we discuss strong uniqueness of Dirichlet operators related to stochastic q... more In this survey paper, we discuss strong uniqueness of Dirichlet operators related to stochastic quantization under exponential (and polynomial) interactions in one-dimensional infinite volume based on joint works with Sergio Albeverio and Michael Rockner (Albeverio et al., J Funct Anal 262:602–638, 2012, [4], Kawabi and Rockner, J Funct Anal 242:486–518, 2007, [11]). We also raise an open problem.
Journal of Mathematical Sciences-the University of Tokyo, 2006
In this paper, we establish the Littlewood-Paley-Stein inequality on general metric spaces under ... more In this paper, we establish the Littlewood-Paley-Stein inequality on general metric spaces under a weaker condition than the lower boundedness ofBakry-Emery's Γ 2. We also discuss Riesz transforms. As examples, we deal with diffusion processes on a path space associated with stochastic partial differential equations (SPDEs in short) and a class ofsuperprocesses with immigration. 1. Framework and Results After the Meyer's celebrated work (16), many authors studied the Lit- tlewood-Paley-Stein inequality by a probabilistic approach. Especially, Shigekawa-Yoshida (20) studied it for symmetric diffusion processes on a general state space. In (20), they assumed the existence ofa suitable core A which is not only a ring but also stable under the operation ofthe semigroup and the infinitesimal generator to employ Bakry-Emery's Γ2-method in the proof, and established the Littlewood-Paley-Stein inequality under that Γ2 is bounded from below. However, it is very difficult to check...
In this article we will give a survey on the Laplace-type asymptotics for the laws of solutions o... more In this article we will give a survey on the Laplace-type asymptotics for the laws of solutions of formal Stratonovich-type stochastic differential equations in Banach spaces. The rigorous meaning of the solutions is given by the rough path theory. The key of the proof is the (stochastic) Taylor expansion, which is, in fact, deterministic in this context. The main example we have in mind is the Brownian motion over loop groups.
We consider a quantum field model with exponential interactions on the twodimensional torus, whic... more We consider a quantum field model with exponential interactions on the twodimensional torus, which is called the exp(Φ)2-quantum field model or HøeghKrohn’s model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.
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