Ergodic Properties of the Equilibrium Process Associated with Infinitely Many Markovian Particles

Shiga, Tokuzo; Takahashi, Yōichirō

doi:10.2977/prims/1195192570

Cited by 10 publications

(3 citation statements)

References 3 publications

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“…We should mention that independent infinite particle processes have been studied by many authors, see e.g. [36], but neither in this paper, nor in any other reference we are aware of, it was proved that the process takes values in the configuration space for all values of t > 0.…”

Section: Introductionmentioning

confidence: 69%

The Heat Semigroup on Configuration Spaces

Kondratiev

Lytvynov

Röckner

2003

Publ. Res. Inst. Math. Sci.

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In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural "Riemannian-like" structure of the configuration space Γ X over a complete, connected, oriented, and stochastically complete Riemannian manifold X of infinite volume, the heat semigroup (e −tH Γ ) t∈R + was introduced and studied in [J. Func. Anal. 154 (1998), 444-500]. Here, H Γ is the Dirichlet operator of the Dirichlet form E Γ over the space L 2 (Γ X , π m ), where π m is the Poisson measure on Γ X with intensity m-the volume measure on X. We construct a metric space Γ ∞ that is continuously embedded into Γ X . Under some conditions on the manifold X, we prove that Γ ∞ is a set of full π m measure and derive an explicit formula for the heat semigroup: (e −tH Γ F )(γ) = Γ∞ F (ξ) P t,γ (dξ), where P t,γ is a probability measure on Γ ∞ for all t > 0, γ ∈ Γ ∞ . The central results of the paper are two types of Feller properties for the heat semigroup. The first one is a kind of strong Feller property with respect to the metric on the space Γ ∞ . The second one, obtained in the case X = R d , is the Feller property with respect to the intrinsic metric of the Dirichlet form E Γ . Next, we give a direct construction of the independent infinite particle process on the manifold X, which is a realization of the Brownian motion on the configuration space. The main point here is that we prove that this process can start in every γ ∈ Γ ∞ , will never leave Γ ∞ , and has continuous sample path in Γ ∞ , provided dim X ≥ 2. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the P t,γ (•) above. Furthermore, we discuss the necessary changes to be done for constructing the process in the case dim X = 1. Finally, as an easy consequence we get a "path-wise" construction of the independent particle process on Γ ∞ from the underlying Brownian motion.

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Section: Introductionmentioning

confidence: 69%

The Heat Semigroup on Configuration Spaces

Kondratiev

Lytvynov

Röckner

2003

Publ. Res. Inst. Math. Sci.

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“…The dynamics of infinitely many stochastic particle has a long a rich history, dating back to Dobrushin [5]. The ergodic properties of independent particles have been early discussed in [15]. More recently, an approach based on Dirichlet forms has been introduced in [1] and successively extensively developed.…”

Section: Introductionmentioning

confidence: 99%

Boundary driven Brownian gas

Bertini¹,

Posta²

2019

ALEA

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We consider a gas of independent Brownian particles on a bounded interval in contact with two particle reservoirs at the endpoints. Due to the Brownian nature of the particles, infinitely many particles enter and leave the system in each time interval. Nonetheless, the dynamics can be constructed as a Markov process with continuous paths on a suitable space. If λ 0 and λ 1 are the chemical potentials of the boundary reservoirs, the stationary distribution (reversible if and only if λ 0 = λ 1 ) is a Poisson point process with intensity given by the linear interpolation between λ 0 and λ 1 . We then analyze the empirical flow that it is defined by counting, in a time interval [0, t], the net number of particles crossing a given point x. In the stationary regime we identify its statistics and show that it is given, apart an x dependent correction that is bounded for large t, by the difference of two independent Poisson processes with parameters λ 0 and λ 1 .

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“…After Surgailis' papers, equilibrium independent infinite particle processes have been studied by many authors, see e.g. [22]. However, the problem of identification of allowed initial configurations of the system was not addressed in these papers.…”

Section: Introductionmentioning

confidence: 99%

Non-equilibrium stochastic dynamics in continuum: The free case

Kondratiev¹,

Lytvynov²,

Röckner³

2008

Condens. Matter Phys.

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We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in X such that, with probability one, infinitely many particles will arrive at this set at some time t > 0. We assume that X has infinite volume and, for each α ≥ 1, we consider the set Θ α of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the α-th power of the volume of the set. We find quite general conditions on the process M which guarantee that the corresponding infinite particle process can start at each configuration from Θ α , will never leave Θ α , and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in X), and free Kawasaki dynamics on the configuration space. We also show that if X = R d , then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics. 2000

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Ergodic Properties of the Equilibrium Process Associated with Infinitely Many Markovian Particles

Cited by 10 publications

References 3 publications

The Heat Semigroup on Configuration Spaces

The Heat Semigroup on Configuration Spaces

Boundary driven Brownian gas

Non-equilibrium stochastic dynamics in continuum: The free case

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