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Dyson Brownian motion on a Jordan curve

Vladislav Guskov, Mingchang Liu, Fredrik Viklund

Abstract

Zabrodin recently proposed a generalization of Dyson Brownian motion to a setting where the particles are confined to a smooth Jordan curve in the plane. In this paper, we discuss a rigorous construction of such a process on a rectifiable Jordan curve and study some of its basic properties. Under further smoothness assumptions, we derive the associated Fokker-Planck-Kolmogorov equation, prove convergence towards the stationary Coulomb gas distribution, study large deviations at low temperature, and derive the limiting mean-field McKean--Vlasov equation in the many-particle limit.

Dyson Brownian motion on a Jordan curve

Abstract

Zabrodin recently proposed a generalization of Dyson Brownian motion to a setting where the particles are confined to a smooth Jordan curve in the plane. In this paper, we discuss a rigorous construction of such a process on a rectifiable Jordan curve and study some of its basic properties. Under further smoothness assumptions, we derive the associated Fokker-Planck-Kolmogorov equation, prove convergence towards the stationary Coulomb gas distribution, study large deviations at low temperature, and derive the limiting mean-field McKean--Vlasov equation in the many-particle limit.
Paper Structure (23 sections, 26 theorems, 218 equations)

This paper contains 23 sections, 26 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Discussion
  5. Existence
  6. Strong solution
  7. Strong local solution
  8. Strong global solution
  9. Back to the curve
  10. Transition probability functions
  11. Fokker-Planck-Kolmogorov equation
  12. Parametrization process
  13. Quotient parametrization process
  14. Convergence to stationary distribution
  15. Poincaré type inequality
  16. ...and 8 more sections

Key Result

Theorem 1.5

Let $\Gamma$ be a rectifiable Jordan curve. For any $\beta\ge 1$ and any collection $\boldsymbol{z} =\{z_{1},\dots, z_{N}\}$ of points on $\Gamma$, ordered counterclockwise, such that $z_{i}\neq z_{j}$, there exists a parametrization process $X = (X_{1}(t), \dots,X_{N}(t))_{t\ge 0}$, with $z_{i} = \ where $B = (B_1(t), \ldots, B_N(t))_{t \ge 0}$ is $N$-dimensional standard Brownian motion. Moreove

Theorems & Definitions (65)

  • Definition 1.1: Parametrization process
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4: Dyson Brownian motion on a Jordan curve
  • Theorem 1.5: Existence for $\beta \ge 1$
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10: Fokker-Planck-Kolmogorov equation
  • ...and 55 more