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

Abstract

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

On non-equilibrium stochastic dynamics for interacting particle systems in continuum

“…Now we would like to list some facts the proofs of which are completely analogous to those proposed in [13,17]. Remark 4.8.…”

“…To this class belongs the Glauber type dynamics in continuum which are under active consideration, see [3,29]. Another class of interesting stochastic processes is formed by the Kawasaki type dynamics in continuum [19] and gradient diffusions [1,17]. Most of the results we have up to now for these processes are related to the equilibrium case (via the Dirichlet forms approach) [13,18] or to the processes in bounded domains (see, e.g., [8,24]).…”

On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum

“…Then, using (8), we see that (17) and (22) converge to (14), whereas (21) and (23) converge to (15). Therefore, (12) and (13) hold.…”

Equilibrium Diffusion on the Cone of Discrete Radon Measures

“…Using the methods of [14,Section 4] (see also [11,Section 6]), one can show that d V (·, γ) ∈ D(E Γ ) and there exists G 1 ∈ L 1 (Γ f ( X), ρ) (independent of γ) such that S Γ (d V (·, γ)) ≤ G 1 ρ-a.e. Hence, we only need to prove that d f (·, γ) ∈ D(E Γ ) and there exists G 2 ∈ L 1 (Γ f ( X), ρ) (independent of γ) such that S Γ (d f (·, γ)) ≤ G 2 ρ-a.e.…”

“…The diffusion process has µ as invariant (and even symmetrizing) measure. To prove the main result, we use the general theory of Dirichlet forms [13] as well as the theory of Dirichlet forms over configuration spaces [14,18], see also [1,11].…”