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Matrix liberation process III: Unitary Brownian motion and martingale analysis

Yoshimichi Ueda

Abstract

We investigate the rate functions that emerge in our previous works towards large deviation principle for the matrix liberation process driven by the unitary Brownian motion as well as the unitary Brownian motion itself. Our approach is grounded in the viewpoint of the martingale problem. Specifically, we formulate and solve a "free martingale problem" within this framework, which provides a new perspective on the underlying stochastic structure.

Matrix liberation process III: Unitary Brownian motion and martingale analysis

Abstract

We investigate the rate functions that emerge in our previous works towards large deviation principle for the matrix liberation process driven by the unitary Brownian motion as well as the unitary Brownian motion itself. Our approach is grounded in the viewpoint of the martingale problem. Specifically, we formulate and solve a "free martingale problem" within this framework, which provides a new perspective on the underlying stochastic structure.

Paper Structure

This paper contains 17 sections, 35 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on unitary processes
  3. Noncommutative coordinates
  4. Derivations and cyclic derivatives
  5. Non-commutative $L^2$- and $L^\infty$-spaces
  6. Continuous tracial states
  7. Rate function
  8. Study on Rate Function
  9. A free martingale problem
  10. The Brownian motion on the unitary group
  11. Smooth continuous tracial states
  12. A weak large deviation result
  13. The (matrix) liberation process
  14. Review on our previous works
  15. Study on the rate function(s)
  16. ...and 2 more sections

Key Result

Lemma 3.1

For any word $w$ in the variables $x_j$ and $u_i(t), u_i(t)^*$, we have: and If $w$ does not contain any letters $u_i(t'), u_i(t')^*$ with $t' \geq t$, then $\mathfrak{D}_{t,i}w = 0$.

Theorems & Definitions (70)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Lemma 3.6
  • ...and 60 more