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This paper is part of a series aiming at proving that the $\limsup$ and$\liminf$ variants of Voiculescu's free entropy coincide. This is based on aLaplace principle (implying a large deviation principle) for hermitian brownianmotion on $[0,1]$. In the current paper, 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 aSchwinger-Dyson equation for subquadratic, bounded below, strictly convexpotentials with Lipschitz derivative sufficiently approximable bynon-commutative polynomials. Our results are based on Dupuis-Ellis weakconvergence approach to large deviations where one shows a Laplace principle inobtaining a stochastic control formulation for exponential functionals. In thenon-commutative context, ultrapoduct analysis replaces weak-convergence of thestochastic control problems.

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