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Mean-Field Limits for Stochastic Interacting Particles on Digraph Measures

Christian Kuehn, Carlos Pulido

Abstract

Many natural phenomena are effectively described by interacting particle systems, which can be modeled using either deterministic or stochastic differential equations (SDEs). In this study, we specifically investigate particle systems modeled by SDEs, wherein the mean field limit converges to a Vlasov-Fokker-Planck-type equation. Departing from conventional approaches in stochastic analysis, we explore the network connectivity between particles using diagraph measures (DGMs). DGMs are one possible tool to capture sparse, intermediate and dense network/graph interactions in the mean-field thereby going beyond more classical approaches such as graphons. Since the main goal is to capture large classes of mean-field limits, we set up our approach using measure-theoretic arguments and combine them with suitable moment estimates to ensure approximation results for the mean-field.

Mean-Field Limits for Stochastic Interacting Particles on Digraph Measures

Abstract

Many natural phenomena are effectively described by interacting particle systems, which can be modeled using either deterministic or stochastic differential equations (SDEs). In this study, we specifically investigate particle systems modeled by SDEs, wherein the mean field limit converges to a Vlasov-Fokker-Planck-type equation. Departing from conventional approaches in stochastic analysis, we explore the network connectivity between particles using diagraph measures (DGMs). DGMs are one possible tool to capture sparse, intermediate and dense network/graph interactions in the mean-field thereby going beyond more classical approaches such as graphons. Since the main goal is to capture large classes of mean-field limits, we set up our approach using measure-theoretic arguments and combine them with suitable moment estimates to ensure approximation results for the mean-field.
Paper Structure (9 sections, 9 theorems, 100 equations)

This paper contains 9 sections, 9 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Metric Measure Spaces and Digraph Measures
  3. Main Results
  4. Assumptions ($H$)
  5. Assumptions ($\tilde{H}$)
  6. Existence of Solutions
  7. Proof of Theorem \ref{['teorema']}
  8. Proof of Theorem \ref{['teorema2']}
  9. Examples

Key Result

Proposition 1

Let $I$ be a complete separable metric space. Assume $I$ is compact. Then $(\mathcal{B}(I, \mathcal{M}_+(I)), d_\infty)$ and $(\mathcal{C}(I, \mathcal{M}_+(I)), d_\infty)$ are complete metric spaces.

Theorems & Definitions (19)

  • Proposition 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • proof
  • proof
  • Proposition 7
  • proof
  • Proposition 8
  • ...and 9 more