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Analysis and geometry on configuration spaces: The Gibbsian case
, 1998
"... Using a natural "Riemanniangeometrylike" structure on the configuration space \Gamma over IR d , we prove that for a large class of potentials OE the corresponding canonical Gibbs measures on \Gamma can be completely characterized by an integration by parts formula. That is, if r \Gamma is th ..."
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Cited by 10 (2 self)
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Using a natural "Riemanniangeometrylike" structure on the configuration space \Gamma over IR d , we prove that for a large class of potentials OE the corresponding canonical Gibbs measures on \Gamma can be completely characterized by an integration by parts formula. That is, if r \Gamma is the gradient of the Riemannian structure on \Gamma one can define a corresponding divergence div \Gamma OE such that the canonical Gibbs measures are exactly those measures ¯ for which r \Gamma and div \Gamma OE are dual operators on L 2 (\Gamma; ¯). One consequence is that for such ¯ the corresponding Dirichlet forms E \Gamma ¯ are defined. In addition, each of them is shown to be associated with a conservative diffusion process on \Gamma with invariant measure ¯. The corresponding generators are extensions of the operator \Delta \Gamma OE := div \Gamma OE r \Gamma . The diffusions can be characterized in terms of a martingale problem and they can be considered as a Brown...
Scaling limit of stochastic dynamics in classical continuous systems
, 2002
"... We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on R d, d ≥ 1. The aim is to derive macroscopic quantities from a given micro or mesoscopic system. The scaling we ..."
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Cited by 5 (5 self)
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We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on R d, d ≥ 1. The aim is to derive macroscopic quantities from a given micro or mesoscopic system. The scaling we
Limit theorems for tagged particles
 Markov Processes Relat. Fields
, 1996
"... Abstract. We review old and new results about the limiting behaviour of a tagged particle in different interacting particle systems: (a) independent particles with no mass in one dimension with continuous paths like Brownian motions and ideal gases; (b) reversible processes on R d or Z d like intera ..."
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Cited by 5 (0 self)
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Abstract. We review old and new results about the limiting behaviour of a tagged particle in different interacting particle systems: (a) independent particles with no mass in one dimension with continuous paths like Brownian motions and ideal gases; (b) reversible processes on R d or Z d like interacting Brownian motions and Kawasaki dynamics; (c) simple exclusion processes; (d) zero range processes; (e) conservative nearest particle processes including the Hammersley process. In each case we consider as initial distribution the invariant distribution for the process as seen from the tagged particle. We review two kinds of limiting behaviour: the law of large numbers and the invariance principle. Denoting by X(t) the position of the tagged particle, the law of large numbers says that as t → ∞, X(t)/t converges almost surely to a constant. The invariance principle means that when conveniently centered and rescaled, the process converges to Brownian motion. When the initial distribution of the particles is not invariant, a weak form of the law of large numbers has been obtained in some cases.
Infinite interaction diffusion particles I: Equilibrium process and its scaling limit
 Forum Math
"... 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 Rd and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y) ..."
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Cited by 4 (3 self)
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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 Rd 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 socalled 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 EΓ µ on L2 (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 EΓ µ on the set of smooth cylinder functions. We then show that, for any Dirichlet form EΓ µ, there exists a diffusion process that is properly associated with it. Finally, in a way parallel to [17], we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C([0, ∞), D ′), where D ′ is the dual space of D:=C ∞ 0 (Rd).
N/VLIMIT FOR STOCHASTIC DYNAMICS IN CONTINUOUS PARTICLE SYSTEMS
, 2005
"... Abstract. We provide an N/Vlimit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on R d, d ≥ 1. Starting point is an Nparticle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ ⊂ R ..."
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Cited by 2 (2 self)
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Abstract. We provide an N/Vlimit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on R d, d ≥ 1. Starting point is an Nparticle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ ⊂ R d with finite volume (Lebesgue measure) V = Λ  < ∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above Nparticle dynamic in Λ as N → ∞ and V → ∞ such that N/V → ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential selfadjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials φ of Ruelle type and all temperatures, densities, and dimensions d ≥ 1, respectively. φ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a byproduct an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition. 1.
Tagged particle process in continuum with singular interactions (preprint) available at arXiv:0804.4868v3
"... Abstract. 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 equations ∞X dξ(t) = ∇φ(yi(t))dt + √ 2 dB1(t), t ≥ 0, (TP) dyi(t) = − ..."
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Cited by 2 (0 self)
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Abstract. 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 equations ∞X dξ(t) = ∇φ(yi(t))dt + √ 2 dB1(t), t ≥ 0, (TP) dyi(t) = − i=1 P∞ j=1 j=i ∇φ(yi(t) − yj(t)) − ∇φ(yi(t)) − P∞ j=1 ∇φ(yj(t)) + √ 2 d(Bi+1(t) − B1(t)), t ≥ 0
Diffusion approximation for equilibrium Kawasaki dynamics in continuum
, 2007
"... A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in R d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure µ as invariant measure. We study a diffusive limit of such a dynamics, derive ..."
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A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in R d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure µ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time. Under weak assumptions on the potential of pair interaction, φ, (in particular, admitting a singularity of φ at zero), we prove that, on a set of smooth local functions, the generator of the scaled dynamics converges to the generator of the gradient stochastic dynamics. If the set on which the generators converge is a core for the diffusion generator, the latter result implies the weak convergence of finitedimensional distributions of the corresponding equilibrium processes. In particular, if the potential φ is from C 3 b (Rd) and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [Choi et al., J. Math.
Alea 7, 193–205 (2010) Crowding of Brownian spheres
"... Abstract. We study two models consisting of reflecting onedimensional Brownian “particles ” of positive radius. We show that the stationary empirical distributions for the particle systems do not converge to the harmonic function for the generator of the individual particle process, unlike in the c ..."
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Abstract. We study two models consisting of reflecting onedimensional Brownian “particles ” of positive radius. We show that the stationary empirical distributions for the particle systems do not converge to the harmonic function for the generator of the individual particle process, unlike in the case when the particles are infinitely small. 1.