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The fluctuation behaviour of the stochastic point vortex model with common noise

Yufei Shao, Xianliang Zhao

TL;DR

This work analyzes fluctuations around the mean-field limit for the stochastic point vortex model on $\mathbb{T}^2$ with common environmental noise. By combining the relative-entropy method with a localization argument, the authors prove that the fluctuation processes converge in distribution to the unique probabilistically strong solution of a linear fluctuation SPDE, establishing strong convergence to the conditional McKean–Vlasov equation driven by the environment. They develop tightness via entropy bounds, identify the limit through strong convergence to the conditional MV dynamics, and prove well-posedness and pathwise uniqueness for the fluctuation SPDE, thereby extending Gaussian fluctuation results to systems with common noise and singular Biot–Savart interactions. The results provide a rigorous central-limit-type description of the stochastic point vortex system under environmental influences, with implications for propagation of chaos under common noise and stochastic fluid models.

Abstract

This article studies the fluctuation behaviour of the stochastic point vortex model with common noise. Using the martingale method combined with a localization argument, we prove that the sequence of fluctuation processes converges in distribution to the unique probabilistically strong solution of a linear stochastic evolution equation. In particular, we establish the strong convergence from the stochastic point vortex model with common noise to the conditional McKean Vlasov equation.

The fluctuation behaviour of the stochastic point vortex model with common noise

TL;DR

This work analyzes fluctuations around the mean-field limit for the stochastic point vortex model on with common environmental noise. By combining the relative-entropy method with a localization argument, the authors prove that the fluctuation processes converge in distribution to the unique probabilistically strong solution of a linear fluctuation SPDE, establishing strong convergence to the conditional McKean–Vlasov equation driven by the environment. They develop tightness via entropy bounds, identify the limit through strong convergence to the conditional MV dynamics, and prove well-posedness and pathwise uniqueness for the fluctuation SPDE, thereby extending Gaussian fluctuation results to systems with common noise and singular Biot–Savart interactions. The results provide a rigorous central-limit-type description of the stochastic point vortex system under environmental influences, with implications for propagation of chaos under common noise and stochastic fluid models.

Abstract

This article studies the fluctuation behaviour of the stochastic point vortex model with common noise. Using the martingale method combined with a localization argument, we prove that the sequence of fluctuation processes converges in distribution to the unique probabilistically strong solution of a linear stochastic evolution equation. In particular, we establish the strong convergence from the stochastic point vortex model with common noise to the conditional McKean Vlasov equation.
Paper Structure (12 sections, 19 theorems, 100 equations)

This paper contains 12 sections, 19 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Related literatures
  3. Outline of proof and difficulties
  4. Preliminaries
  5. Definitions of solutions
  6. Relative entropy and stochastic point vortex model
  7. Besov space
  8. Uniform estimates
  9. Estimates on the relative entropy
  10. Tightness
  11. Well posedness of the fluctuation SPDE
  12. Identification of the limiting points

Key Result

Theorem 1.1

Assume that the identical distribution $\mathcal{L}(X(0))$ on torus $\mathbb{T}^2$ for the i.i.d. initial values $\{X^N_i(0),i\in \mathbb{N}\}$ has a density $v_0\in H^{3}(\mathbb{T}^2)$ with strictly positive lower bound, i.e., $\inf_{\mathbb{T}^2}v_0>0.$ the sequence of fluctuation measures $\eta^ for every $\alpha>1$, where $\eta$ is the unique probabilistically strong solution to the fluctuati

Theorems & Definitions (33)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8: jabin2018quantitative
  • ...and 23 more