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Orthogonal Intertwiners for Infinite Particle Systems in The Continuum

Stefan Wagner

Abstract

This article focuses on a system of sticky Brownian motions, also known as Howitt-Warren martingale problem, and correlated Brownian motions and shows that infinite-dimensional orthogonal polynomials intertwine the dynamics of infinitely many particles and their $n$-particle evolution. The proof is based on two assumptions about the model: information about the reversible measures for the $n$-particle dynamics and consistency. Additionally, explicit formulas for the polynomials are used, including a new explicit formula for infinite-dimensional Meixner polynomials, the orthogonal polynomials with respect to the Pascal process. As an application of the intertwining relations, new reversible measures for the infinite-particle dynamics are obtained.

Orthogonal Intertwiners for Infinite Particle Systems in The Continuum

Abstract

This article focuses on a system of sticky Brownian motions, also known as Howitt-Warren martingale problem, and correlated Brownian motions and shows that infinite-dimensional orthogonal polynomials intertwine the dynamics of infinitely many particles and their -particle evolution. The proof is based on two assumptions about the model: information about the reversible measures for the -particle dynamics and consistency. Additionally, explicit formulas for the polynomials are used, including a new explicit formula for infinite-dimensional Meixner polynomials, the orthogonal polynomials with respect to the Pascal process. As an application of the intertwining relations, new reversible measures for the infinite-particle dynamics are obtained.
Paper Structure (18 sections, 15 theorems, 75 equations)

This paper contains 18 sections, 15 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Correlated Brownian Motions
  3. The Model
  4. Unlabeled Dynamics
  5. Multiple Wiener-Itô Integrals for the Poisson Process
  6. Main Result
  7. Sticky Brownian Motions
  8. The Model
  9. Pascal Process and Infinite-Dimensional Meixner Polynomials
  10. Main Result
  11. Strategy for the Proof
  12. Consistency
  13. Main Result: a Broader Perspective
  14. Proofs
  15. Unlabeled Dynamics and Consistency
  16. ...and 3 more sections

Key Result

Lemma 2.2

For each $0 \leq a \leq 1$, there exists a Markov family $(\Omega$, $\mathcal{F}$, $(\eta_t)_{t \geq 0}$, $(\mathbb{P}_\mu)_{\mu \in \mathbf{N}})$ with state space $\mathbf{N}$ such that $\eta_t$, $t \geq 0$ is proper and describes the evolution of an unlabeled system of correlated Brownian motions

Theorems & Definitions (36)

  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Corollary 2.6
  • proof
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • ...and 26 more