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Long Time Entropy-Cost type Propagation of Chaos

Xing Huang

TL;DR

This work addresses the long-time propagation of chaos for mean-field interacting particle systems with stochastic noise by introducing an entropy-cost type propagation bound that links the entropy of finite-particle marginals to a weaker initial $L^2$-Wasserstein distance. A general theorem converts finite-time entropy estimates and uniform-in-time Wasserstein contraction into explicit long-time bounds, enabling entropy-controlled chaos for large systems. The paper then applies these results to path-independent models with multiplicative noise, obtaining explicit exponential decay rates and invariant-measure bounds, and extends to path-dependent models with additive noise, including both non-degenerate and degenerate diffusion regimes, with precise rates and projection/invariant-measure conclusions even when log-Sobolev inequalities fail. Overall, the results provide a robust, quantitative framework for long-time chaos in a broad class of mean-field dynamics, encompassing dissipative, path-dependent, and degenerate settings.

Abstract

Due to the regularization effect of the stochastic noise, the quantitative entropy-cost type propagation of chaos for mean field interacting particle system is proposed. The result shows that the Kac's chaotic property measured in relative entropy at any positive time can only depend on the weaker initial one measured in $L^2$-Wasserstein distance. Moreover, under dissipative assumption, the long time entropy-cost type propagation of chaos can also be captured. The results are also available in path dependent case, where the log-Sobolev inequality for McKean-Vlasov SDEs does not hold.

Long Time Entropy-Cost type Propagation of Chaos

TL;DR

This work addresses the long-time propagation of chaos for mean-field interacting particle systems with stochastic noise by introducing an entropy-cost type propagation bound that links the entropy of finite-particle marginals to a weaker initial -Wasserstein distance. A general theorem converts finite-time entropy estimates and uniform-in-time Wasserstein contraction into explicit long-time bounds, enabling entropy-controlled chaos for large systems. The paper then applies these results to path-independent models with multiplicative noise, obtaining explicit exponential decay rates and invariant-measure bounds, and extends to path-dependent models with additive noise, including both non-degenerate and degenerate diffusion regimes, with precise rates and projection/invariant-measure conclusions even when log-Sobolev inequalities fail. Overall, the results provide a robust, quantitative framework for long-time chaos in a broad class of mean-field dynamics, encompassing dissipative, path-dependent, and degenerate settings.

Abstract

Due to the regularization effect of the stochastic noise, the quantitative entropy-cost type propagation of chaos for mean field interacting particle system is proposed. The result shows that the Kac's chaotic property measured in relative entropy at any positive time can only depend on the weaker initial one measured in -Wasserstein distance. Moreover, under dissipative assumption, the long time entropy-cost type propagation of chaos can also be captured. The results are also available in path dependent case, where the log-Sobolev inequality for McKean-Vlasov SDEs does not hold.
Paper Structure (6 sections, 5 theorems, 120 equations)

This paper contains 6 sections, 5 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. A general result
  3. Applications in path independent case with multiplicative noise
  4. Application in path dependent models with additive noise
  5. The case with non-degenerate diffusion coefficients
  6. The case with degenerate diffusion coefficients

Key Result

Theorem 2.1

Assume that there exist constants $t_0>r_0\geq 0, c>0$ and continuous functions $g:[0,\infty)\rightarrow(0,\infty), h:(r_0,t_0]\rightarrow(0,\infty)$ with $\lim_{t\rightarrow\infty} g(t)=0, \lim_{t\rightarrow r_0} h(t)=\infty$ such that for any $N\geq 1$ and $\mu\in\mathscr P_2(E)$, and Then the following assertions hold.

Theorems & Definitions (13)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['POC30']}
  • Remark 3.5
  • Theorem 4.1
  • ...and 3 more