Bulk Diffusion for Interacting Brownian Particles

Guo, Maozheng; Papanicolaou, George

doi:10.1007/978-1-4899-6653-7_3

Cited by 18 publications

(23 citation statements)

References 1 publication

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“…We shall now discuss a scaling limit of the diffusion process constructed in Theorem 6.1, the scaling being absolutely analogous to the one considered in [9,39,44,18,17].…”

Section: Scaling Limit Of the Stochastic Dynamicsmentioning

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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“…We shall now discuss a scaling limit of the diffusion process constructed in Theorem 6.1, the scaling being absolutely analogous to the one considered in [9,39,44,18,17].…”

Section: Scaling Limit Of the Stochastic Dynamicsmentioning

confidence: 99%

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

2006

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“…Neither Theorem 1.3, nor the tightness results of Section 2 are restricted to SSEP. Theorem 1.3 has been shown for so-called interacting difusions by Guo-Papanicolaou (see [3] and also [I]), and a generalization of Theorem 1.3 to the class of symmetric simple exclusion processes with speed change can be found in 181.…”

Section: Suppose ( P ( Z )mentioning

confidence: 99%

Propagation of chaos for symmetric simple exclusions

Rezakhanlou

1994

Comm Pure Appl Math

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Notation and SummaryThe symmetric simple exclusion process (SSEP) is a continuous time Markov process in which particles perform symmetric random walks on a lattice but are excluded to occupy the same site. It is one of the simplest lattice gas models that exhibits some of the behavior expected for a large class of interacting particle systems. The object of this article is to establish propagation of chaos type behavior for SSEP. Roughly speaking, we show that if particles are initially located independently on a lattice, then, despite interaction among them, the independence is restored asymptotically at later times. We assume that initially particles are distributed identically and we show that the evolution of a single marked particle is governed by an inhomogeneous Markov process that can be uniquely determined in terms of the initial distribution and certain transport coeflcients.To define SSEP we have to determine its state space and infinitesimal generator. Let ZL denote the periodic lattice (0, 1,. . . , t} with zero and L identified.We define Zd, as the product of d copies of ZL. Let EL,N denote the space of configurations x = (XI,. . . , x~) with xi E Zf. We think of xi as the location of the i-th particle and we sometimes refer to the index of xi as the label of the particle.Suppose ( p ( z ) : z E Zd) is a symmetric probability density function of finite range; that is to say, c.. For each ( N , L ) , let ,LLN,L be a probability measure on EN,L. Throughout this paper we assume:

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“…Such processes have been studied in e.g. [GP85], [DMFGW89] and [Osa98]. I.e., we consider the heuristic system of stochastic differential equations…”

Section: Tagged Particle Process In Continuum With Singular Interactimentioning

confidence: 99%

Tagged Particle Process in Continuum With Singular Interactions

Fattler

Grothaus

2011

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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We study the dynamics of a tagged particle in an infinite particle environment. Such processes have been studied in e.g. [GP85], [DMFGW89] and [Osa98]. I.e., we consider the heuristic system of stochastic differential equationsThis system realizes the coupling of the motion of the tagged particle, described by (TP), and the motion of the environment seen from the tagged particle, described by (ENV). As we can observe in (TP) the solution to (ENV), the so-called environment process, is driving the tagged particle. Thus our strategy is to study (ENV) at first and afterwards the coupled process, i.e., (TP) and (ENV) simultaneously. Here the analysis and geometry on configuration spaces developed in [AKR98a] and [AKR98b] plays an important role. Furthermore, the harmonic analysis on configuration spaces derived in [KK02] is very useful for our considerations.First we derive an integration by parts formula with respect to the standard gradient ∇ Γ on configuration spaces Γ for a general class of grand canonical Gibbs measures µ, corresponding to pair potentials φ and intensity measures σ = z exp(−φ) dx, 0 < z < ∞, having correlation functions fulfilling a Ruelle bound. Furthermore, we use a second integration by parts formula with respect to the gradient ∇ Γ γ , generating the uniform translations on Γ, for a (non-empty) subclass of the Gibbs measures µ as above which is provided in [CK09]. Combining these two gradients by Dirichlet form techniques we can construct the environment process and the coupled process, respectively. Scaling limits of such dynamics have been studied e.g. in [DMFGW89], [GP85] and [Osa98]. Our results give the first mathematically rigorous and complete construction of the tagged particle process in continuum with interaction potential. In particular, we can treat interaction potentials which might have a singularity at the origin, non-trivial negative part and infinite range as e.g. the Lennard-Jones potential.

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Bulk Diffusion for Interacting Brownian Particles

Cited by 18 publications

References 1 publication

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

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

Propagation of chaos for symmetric simple exclusions

Tagged Particle Process in Continuum With Singular Interactions

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