A Laplace Principle for Hermitian Brownian Motion and Free Entropy I: the convex functional case

Abstract

This paper is part of a series aiming at proving that the $\mathrm{limsup}$ and $\mathrm{liminf}$ variants of Voiculescu's free entropy coincide. This is based on a Laplace principle (implying a large deviation principle) for hermitian brownian motion on $[0,1]$. In the current paper, we show that microstates free entropy $\chi (X_1,...,X_m)$ and non-microstate free entropy ${\chi }^{\ast }(X_1,...,X_m)$ coincide for self-adjoint variables $(X_1,...,X_m)$ satisfying a Schwinger-Dyson equation for subquadratic, bounded below, strictly convex potentials with Lipschitz derivative sufficiently approximable by non-commutative polynomials. Our results are based on Dupuis-Ellis weak convergence approach to large deviations where one shows a Laplace principle in obtaining a stochastic control formulation for exponential functionals. In the non-commutative context, ultrapoduct analysis replaces weak-convergence of the stochastic control problems.

83 pages. Major revision: First part of a revised corrected and expanded version now split in two papers. It contains the most probabilistic part of the two. Partially new presentation with better emphasis on technical tools used : control, FBSDE (with a new section previously hidden in a long and technical proof)

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