Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions

Osada, Hirofumi

doi:10.1007/bf02099365

Cited by 97 publications

(133 citation statements)

References 13 publications

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“…(v) Spohn [38] constructed nonintersecting Brownian motions on a torus and discussed the infinite volume limit to an infinite system of Dyson-type Brownian motions, which was also constructed by Dirichlet form technique in Osada [32]. For the present N nonintersecting Brownian motion X(t) in a finite time interval (0, T ], two types of temporally inhomogeneous infinite particle systems are obtained by setting T = T (N ) and taking N → ∞.…”

Section: Remarksmentioning

confidence: 99%

Functional central limit theorems for vicious walkers

Katori

Tanemura

2003

Stochastics and Stochastic Reports

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Dedicated to Professor Tokuzo Shiga on his 60th birthday.Abstract. We consider the diffusion scaling limit of the vicious walker model that is a system of nonintersecting random walks. We prove a functional central limit theorem for the model and derive two types of nonintersecting Brownian motions, in which the nonintersecting condition is imposed in a finite time interval (0, T ] for the first type and in an infinite time interval (0, ∞) for the second type, respectively. The limit process of the first type is a temporally inhomogeneous diffusion, and that of the second type is a temporally homogeneous diffusion that is identified with a Dyson's model of Brownian motions studied in the random matrix theory. We show that these two types of processes are related to each other by a multi-dimensional generalization of Imhof's relation, whose original form relates the Brownian meander and the three-dimensional Bessel process. We also study the vicious walkers with wall restriction and prove a functional central limit theorem in the diffusion scaling limit.

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Section: Remarksmentioning

confidence: 99%

Functional central limit theorems for vicious walkers

Katori

Tanemura

2003

Stochastics and Stochastic Reports

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“…It is known [12] that X is a H-valued diusion process with uncountable invariant probability measures (equilibrium distributions). Let l z denote translation invariant grand canonical (g.c.)…”

Section: And 1x2mentioning

confidence: 99%

Positivity of the self-diffusion matrix of interacting Brownian particles with hard core

Osada

1998

Probab Theory Relat Fields

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We prove the positivity of the self-diusion matrix of interacting Brownian particles with hard core when the dimension of the space is greater than or equal to 2. Here the self-diusion matrix is a coecient matrix of the diusive limit of a tagged particle. We will do this for all activities, z b 0, of Gibbs measures; in particular, for large z ± the case of high density particles. A typical example of such a particle system is an in®nite amount of hard core Brownian balls.Mathematics Subject Classi®cation (1991): 60K35, 60J60, 82C2

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“…The study of such diffusions has been initiated by R. Lang [24] (see also [42,13]), who considered the case φ ∈ C 3 0 (R d ) using finite-dimensional approximations of stochastic differential equations. More singular φ, which are of particular interest from the point of view of statistical mechanics, have been treated by H. Osada [32] and M. Yoshida [46]. These authors were the first to use the Dirichlet form approach from [27] for the construction of such processes.…”

Section: Introductionmentioning

confidence: 99%

Infinite interacting diffusion particles I: Equilibrium process and its scaling limit

2006

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A stochastic dynamics (X(t)) t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ of all locally finite subsets (configurations) in R d and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y). An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics-the so-called gradient stochastic dynamics, or interacting Brownian particles-has been investigated. By using the theory of Dirichlet forms from [27], we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form E Γ µ on L 2 (Γ; µ), and under general conditions on the potential φ, prove its closability. For a potential φ having a "weak" singularity at zero, we also write down an explicit form of the generator of E Γ µ on the set of smooth cylinder functions. We then show that, for any Dirichlet form E

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Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions

Cited by 97 publications

References 13 publications

Functional central limit theorems for vicious walkers

Functional central limit theorems for vicious walkers

Positivity of the self-diffusion matrix of interacting Brownian particles with hard core

Infinite interacting diffusion particles I: Equilibrium process and its scaling limit

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