Stochastic differential equations in a scale of Hilbert spaces 2. Global solutions

TL;DR

This paper extends the theory of stochastic differential equations in Hilbert spaces by proving existence and uniqueness of infinite-time solutions, and applies these results to model non-equilibrium stochastic dynamics of infinite particle systems.

Contribution

It advances previous work by establishing global solutions for stochastic differential equations in a scale of Hilbert spaces, enabling analysis of infinite-time behavior.

Findings

Proved existence and uniqueness of global solutions.

Applied results to non-equilibrium dynamics of infinite particle systems.

Improved upon finite-time solution results from prior research.

Abstract

A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence, uniqueness and path-continuity of infinite-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 ${\mathbb{R}}^{n}$. The paper improves the results of the work by the second named author "Stochastic differential equations in a scale of Hilbert spaces", Electron. J. Probab. 23, where finite-time solutions were constructed.