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Quantitative propagation of chaos for 2D stochastic vortex model on the whole space under moderate interactions

Alexandre B. de Souza

Abstract

We derive the stochastic 2D vortex model on the whole Euclidean space from stochastic particle systems driven by individual and environmental noises, obtaining pathwise quantitative bounds in the sense of relative entropy. The main novelty is the application of the Donsker-Varadhan inequality in the context of moderately interacting particles to handle the nonlinearity, as well as the use of localization techniques combined with the probabilistic data setting, to derive estimates for the quadratic variation terms. Moreover, we prove the existence of a suitable solution to the aforementioned model.

Quantitative propagation of chaos for 2D stochastic vortex model on the whole space under moderate interactions

Abstract

We derive the stochastic 2D vortex model on the whole Euclidean space from stochastic particle systems driven by individual and environmental noises, obtaining pathwise quantitative bounds in the sense of relative entropy. The main novelty is the application of the Donsker-Varadhan inequality in the context of moderately interacting particles to handle the nonlinearity, as well as the use of localization techniques combined with the probabilistic data setting, to derive estimates for the quadratic variation terms. Moreover, we prove the existence of a suitable solution to the aforementioned model.
Paper Structure (15 sections, 5 theorems, 91 equations)

This paper contains 15 sections, 5 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Assumptions
  4. Statement of the main results
  5. Proofs of main results
  6. Time evolution of the relative entropy
  7. Estimates for $II_t + IV_t$: quadratic variation terms
  8. Estimates for $I_t$: the nonlinear terms
  9. Application of Gronwall's Lemma
  10. End of the proof: Removing the stopping time
  11. Appendix
  12. Proof of Theorem \ref{['existence']}
  13. Solving (\ref{['SPDE_Ito']}) for $\sigma=0$
  14. Solving (\ref{['SPDE_Ito']}) for $\sigma \neq 0$
  15. Inequalities

Key Result

Theorem 1

Assume $(\mathbf{A}^{\nabla\cdot K})$, $(\mathbf{A}^{\sigma})$, and $(\mathbf{A}^{\rho_0})$. Then, there exist $T_{1} >0$, $C_3 >0$, such that if $\widetilde{C}_3 < C_3$,

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Remark 3
  • proof : Proof of (\ref{['decay rho']})
  • Lemma 3: Csiszár-Kullback-Pinsker inequality
  • Lemma 4: Donsker-Varadhan Inequality