Decay of semigroup for an infinite interacting particle system on continuum configuration spaces

Abstract

We show the heat kernel type variance decay t − d 2 , up to a logarithmic correction, for the semigroup of an infinite particle system on R d , where every particle evolves following a divergence-form operator with diffusivity coefficient that depends on the local configuration of particles. The proof relies on the strategy from [30], and generalizes the localization estimate to the continuum configuration space introduced by S. Albeverio, Y.G. Kondratiev and M. Röckner.

Quantitative homogenization of interacting particle systems

“…Proof. The proof of this lemma borrows some elements from [41,Lemma 4.8]; in both settings, the main point is to construct and analyze an appropriate "cut-off" version of the function u. We use the function Ãs,ε u ∈ H 1 0 (Q s+ε ) defined in eq.…”

“…In relation to the purposes of the present paper, several works considered the problem of obtaining a rate of convergence to equilibrium for a system of interacting particles [54,30,16,43,51,19,41]. Heat kernel bounds for the tagged particle in a simple exclusion process were obtained in [35].…”

Smoothness of the diffusion coefficients for particle systems in continuous space

“…We refer to (3.3) below for the definition of the gradient of a sufficiently smooth function defined on M δ (R d ), and [22] for a rigorous construction of the stochastic process.…”

Quantitative homogenization of interacting particle systems

“…For specific choices of the observable f , the statement of Theorem 1.1 gives an upper bound on the next-order correction to the leading-order behavior of E ρ [(P t f ) 2 ]. For different classes of functions f , the leading-order behavior was investigated in [29] for the model we also consider here. Other models were considered in [7,9,16,31,37,39]; see also [43] that relates this sort of problem with the quantitative homogenization of elliptic equations.…”