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

TL;DR

This paper establishes a decay rate for the heat kernel of an infinite particle system on continuum spaces, extending previous lattice models to more general configuration spaces with configuration-dependent diffusivity.

Contribution

It generalizes localization estimates from lattice models to continuum configuration spaces for infinite particle systems with configuration-dependent diffusion.

Findings

Proves $t^{-d/2}$ decay rate for the semigroup's variance.

Extends localization estimates to continuum configuration spaces.

Adapts strategies from lattice zero range models to continuum settings.

Abstract

We show the heat kernel type variance decay $t^{-\frac{d}{2}}$, up to a logarithmic correction, for the semigroup of an infinite particle system on $\mathbb{R}^d$, 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 $\mathbb{Z}^d$ zero range model, and generalizes the localization estimate to the continuum configuration space introduced by S. Albeverio, Y.G. Kondratiev and M. R\"ockner.