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Multilevel Dyson Brownian motions via the superposition principle

Benjamin Budway, Mykhaylo Shkolnikov

Abstract

Multilevel Dyson Brownian motions (MDBMs) combine Dyson Brownian motions of different dimensions into a single process in a canonical way. This paper completes the theory of MDBMs for $β\ge2$. Specifically, we use the superposition principle of Figalli and Trevisan to construct the MDBMs for all $β>2$ in a unified manner. This also extends their stochastic differential equation representation, first discovered by Gorin and Shkolnikov, to all $β>2$ and proves the uniqueness of the MDBMs for all $β>2$. Finally, we show that their limit as $β\downarrow2$ is given by the $β=2$ MDBM, commonly referred to as the Warren process.

Multilevel Dyson Brownian motions via the superposition principle

Abstract

Multilevel Dyson Brownian motions (MDBMs) combine Dyson Brownian motions of different dimensions into a single process in a canonical way. This paper completes the theory of MDBMs for . Specifically, we use the superposition principle of Figalli and Trevisan to construct the MDBMs for all in a unified manner. This also extends their stochastic differential equation representation, first discovered by Gorin and Shkolnikov, to all and proves the uniqueness of the MDBMs for all . Finally, we show that their limit as is given by the MDBM, commonly referred to as the Warren process.
Paper Structure (10 sections, 19 theorems, 187 equations)

This paper contains 10 sections, 19 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the Dixon-Anderson density
  3. Superposition principle
  4. Summary of the superposition principle
  5. Superposition principle in our context
  6. Proof of Theorem \ref{['MainTheorem']}
  7. Proof of Theorem \ref{['convergence']}
  8. Preliminaries on the Submartingale Problem
  9. Proof of Theorem \ref{['convergence']}
  10. Some Properties of Dyson Brownian Motion

Key Result

Theorem 1.1

Fix a $\theta>1$ and a probability measure $\mu_0$ on the Weyl chamber and for $t>0$ let $\mu_t$ be the time $t$ distribution of a $\theta$-DBM started from $\mu_0$. Then, there exists a $\mathcal{K}$-valued weak solution $X^\theta$ to the SDE MLSDE with the initial distribution $\mu_0 \times \Lambda^{N,\theta}$ such that Moreover, the $\mathcal{K}$-valued weak solution satisfying (b) is unique

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 16 more