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Viscosity Solutions in Non-commutative Variables

Wilfrid Gangbo, David Jekel, Kyeongsik Nam, Aaron Z. Palmer

Abstract

Motivated by parallels between mean field games and random matrix theory, we develop stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equations in the setting of non-commutative variables. Rather than real vectors, the inputs to the equation are tuples of self-adjoint operators from a tracial von Neumann algebra. The individual noise from mean field games is replaced by a free semi-circular Brownian motion, which describes the large-$n$ limit of Brownian motion on the space of self-adjoint matrices. We introduce a classical common noise from mean field games into the non-commutative setting as well, allowing the problems to combine both classical and non-commutative randomness.

Viscosity Solutions in Non-commutative Variables

Abstract

Motivated by parallels between mean field games and random matrix theory, we develop stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equations in the setting of non-commutative variables. Rather than real vectors, the inputs to the equation are tuples of self-adjoint operators from a tracial von Neumann algebra. The individual noise from mean field games is replaced by a free semi-circular Brownian motion, which describes the large- limit of Brownian motion on the space of self-adjoint matrices. We introduce a classical common noise from mean field games into the non-commutative setting as well, allowing the problems to combine both classical and non-commutative randomness.

Paper Structure

This paper contains 46 sections, 34 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Control problems in the multimatrix and von Neumann algebraic settings
  4. Overview of results and organization
  5. Acknowledgements
  6. Preliminaries
  7. Von Neumann Algebras
  8. Free independence and free products
  9. Non-commutative Laws and the weak* topology
  10. Control problem with free individual noise and common noise
  11. Example Problems
  12. Quadratic cost
  13. Eikonal controls
  14. Controlled von Neumann equation
  15. Setup and Assumptions for the General Problem
  16. ...and 31 more sections

Key Result

Lemma 2.1

Let $J$ be any index set. Let $(\mathcal{A}_j)_{j \in J}$ and $\mathcal{B}$ be tracial $\mathrm{W}^*$-algebras, and let $\varphi_j: \mathcal{B} \to \mathcal{A}_j$ be a tracial $W^*$--embedding. Then there exists a tracial $\mathrm{W}^*$-algebra $\mathcal{C}$ and tracial $W^*$--embeddings $\iota_j: \ Moreover, if $\widetilde{\mathcal{C}}$ and $\widetilde{\iota}_j$ are another tracial $\mathrm{W}^*$

Theorems & Definitions (70)

  • Lemma 2.1
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4: Definition 3.5 in gangbo2022duality
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 60 more