# Stochastic differential equations in a scale of Hilbert spaces

## TL;DR

This paper studies stochastic differential equations within a hierarchy of Hilbert spaces, proving existence and uniqueness of solutions, and applies these results to model non-equilibrium stochastic dynamics of infinite particle systems.

## Contribution

It extends the Ovsyannikov method to establish solution existence and uniqueness for equations in a scale of Hilbert spaces, with applications to particle system dynamics.

## Key findings

- Proved existence and uniqueness of solutions in a scale of Hilbert spaces.
- Applied the theoretical results to non-equilibrium stochastic dynamics of infinite particle systems.
- Extended the Ovsyannikov method for this class of stochastic equations.

## 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 a Euclidean space.