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

Giulia Cavagnari, Giuseppe Savaré, Giacomo Enrico Sodini

TL;DR

This work develops a measure-theoretic framework for the convergence of stochastic time-discretization schemes, notably the explicit Euler method, for evolution equations driven by random velocity fields in the space of probability measures. By recasting dynamics as multivalued probability vector fields (MPVFs) on $(oldsymbol{P}_2(oldsymbol{X}), W_2)$ and employing barycentric projections, the authors prove strong convergence of interpolated trajectory laws to the $oldsymbol{\lambda}$-evolution variational inequality (EVI) solution generated by the maximal dissipative extension of the barycenter. They establish lifting results that connect discrete Euler updates with continuous-time curve measures, derive semigroup structures under total dissipativity, and show stability under approximations. The framework is illustrated through stochastic dissipative flows and nonlocal interaction fields, covering stochastic gradient-descent-like dynamics and interacting particle systems, which highlights the approach’s broad applicability to continuous-time stochastic schemes in measure spaces.

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.

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

TL;DR

This work develops a measure-theoretic framework for the convergence of stochastic time-discretization schemes, notably the explicit Euler method, for evolution equations driven by random velocity fields in the space of probability measures. By recasting dynamics as multivalued probability vector fields (MPVFs) on and employing barycentric projections, the authors prove strong convergence of interpolated trajectory laws to the -evolution variational inequality (EVI) solution generated by the maximal dissipative extension of the barycenter. They establish lifting results that connect discrete Euler updates with continuous-time curve measures, derive semigroup structures under total dissipativity, and show stability under approximations. The framework is illustrated through stochastic dissipative flows and nonlocal interaction fields, covering stochastic gradient-descent-like dynamics and interacting particle systems, which highlights the approach’s broad applicability to continuous-time stochastic schemes in measure spaces.

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.

Paper Structure

This paper contains 13 sections, 27 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dissipative Probability Vector Fields and EVI solutions
  4. Explicit Euler scheme and lifting to curves
  5. Total dissipativity and lifting to curves
  6. MPVFs with totally dissipative barycenter
  7. Examples
  8. Stochastic Dissipative Flow
  9. Interaction field
  10. Nonlocal Stochastic Dissipative Flow
  11. Fully stochastic interaction field
  12. Technical results
  13. Sticky particles representation

Key Result

Theorem 2.1

Let $\mathbb X, X$ be Lusin completely regular topological spaces, $\boldsymbol\mu\in\mathcal{P}(\mathbb{X})$, $r:\mathbb{X}\to X$ be a Borel map. Then, there exists a $(r_{\sharp}\boldsymbol\mu)$-a.e. uniquely determined Borel family of probability measures $\{\boldsymbol\mu_x\}_{x\in X}\subset\mat for every bounded Borel map $\varphi:\mathbb{X}\to\mathbb{R}$.

Theorems & Definitions (71)

  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4: Multivalued Probability Vector Field - MPVF
  • Definition 2.5
  • Definition 2.6: Metric-duality pairings
  • Definition 2.7: Metric $\lambda$-dissipativity
  • Definition 2.8: Total $\lambda$-dissipativity
  • Definition 2.9: $\lambda$-Evolution Variational Inequality
  • Definition 3.1: Explicit Euler Scheme
  • ...and 61 more