Large deviations upper bounds for the laws of matrix-valued processes and non-communicative entropies

Using It (o) over caps calculus, we study the large deviations properties of the law of the spectral measure of the Hermitian Brownian motion. We extend this strategy to the symmetric, unitary and Wishart processes. This dynamical approach is generalized to the study of the large deviations of the non-commutative laws of several independent Hermitian Brownian motions. As a consequence, we can bound from above entropies defined in the spirit of the microstates entropy introduced by Voiculescu.