A Laplace Principle for Hermitian Brownian Motion and Free Entropy

We prove that the $limsup$ and $liminf$ variants of Voiculescuu0027s free entropy coincide. This is based on a Laplace principle (implying a full large deviation principle) for hermitian brownian motion on $[0,1]$. As a consequence, we show that microstates free entropy $chi(X_1,...,X_m)$ and non-microstate free entropy $chi^*(X_1,...,X_m)$ coincide for self-adjoint variables $(X_1,...,X_m)$ satisfying a Schwinger-Dyson equation for subquadratic, bounded bellow, strictly convex potentials with Lipschitz derivative sufficiently approximable by non-commutative polynomials. Applying the contraction principle, we obtain a large deviation result for Haar unitaries and deduce the most general additivity property for a new extended definition of orbital free entropy. 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.