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

This paper is part of a series aiming at proving that the $\mathrm{lim\hspace{0.17em}sup}$ and $\mathrm{lim\hspace{0.17em}inf}$ 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 $\chi (X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m)$ and non-microstate free entropy ${\chi }^{\ast }(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m)$ coincide for self-adjoint variables $(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m)$ 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.