Free diffusions, free entropy and free Fisher information

Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX(t) = dS(t) - 1/2 f (X-t) dt where S-t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as t --> infinity, towards the master field (i.e., the Gibbs state). (C) 2001 Editions scientifiques et medicales Elsevier SAS.