  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 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 $\chi^*(X_1,...,X_m)$
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.
