Stochastic Euler Schemes and Dissipative Evolutions in the Space of Probability Measures

TL;DR

This paper investigates the convergence of stochastic discretization schemes for evolution equations in probability measure spaces, providing a unified measure-theoretic framework applicable to various stochastic systems.

Contribution

It introduces a novel framework based on Multivalued Probability Vector Fields to analyze stochastic evolutions in Wasserstein space, establishing convergence under dissipativity conditions.

Findings

Convergence of laws of interpolated trajectories to a limiting evolution.

Applicability to stochastic gradient descent and interacting particle systems.

General measure-theoretic approach for stochastic scheme convergence.

Abstract

We study the convergence of stochastic time-discretization schemes for evolution equations driven by random velocity fields, including examples like stochastic gradient descent and interacting particle systems. Using a unified framework based on Multivalued Probability Vector Fields, we analyze these dynamics at the level of probability measures in the Wasserstein space. Under suitable dissipativity and boundedness conditions, we prove that the laws of the interpolated trajectories converge to those of a limiting evolution governed by a maximal dissipative extension of the associated barycentric field. This provides a general measure-theoretic study for the convergence of stochastic schemes in continuous time.