Stochastic differential equations in a scale of Hilbert spaces

Alexei Daletskii

doi:10.1214/18-ejp247

Abstract

A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in ${\mathbb{R} }^{n}$.

Highlights

Evolution differential and stochastic differential equations in Banach spaces play hugely important role in many parts of mathematics and its applications
The existence, uniqueness and regularity of solutions of the corresponding Cauchy problem can be proved. This classical theory does not cover some important examples motivated by e.g. problems of statistical mechanics and hydrodynamics
It is anticipated that these results can be combined with the approach proposed in the present paper allowing to build stochastic dynamics on the marked configuration space Γ(X, S)

Summary

Introduction

Evolution differential and stochastic differential equations in Banach spaces play hugely important role in many parts of mathematics and its applications. Construction of non-equilibrium stochastic dynamics of infinite particle systems of the aforementioned type has been a long-standing problem, even in the case of linear drift and a single-particle diffusion coefficient It has become important in the framework of analysis on spaces Γ(X, S) of configurations {(x, σx)}x∈γ with marks It is anticipated that (some of) these results can be combined with the approach proposed in the present paper allowing to build stochastic dynamics on the marked configuration space Γ(X, S) Another potential field of applications of the present results is the study of stochastic perturbations of certain (non-local) partial differential equations, cf [4] and [7]. We prove the uniqueness of the infinite-particle dynamics using more classical methods, which generalise those applied to deterministic systems in [24], [9]

Setting

Main results

Stochastic spin dynamics of a quenched particle system

Existence of the dynamics

The uniqueness

Proofs of auxiliary results