Interacting Diffusion Processes

Abstract

The phenomenon ‘propagation of chaos’ is discussed in terms of the relative entropy in (5.26) which allows general microscopical systems and which provides results in the tradition of statistical mechanics. In Section 6.3 we show ‘propagation of chaos in entropy’ for particle clouds with prescribed initial and final distributions: The particles become asymptotically independent and perform identically according to a Csiszar projection as their number increases to infinity. These limiting distributions turn out to be Schrödinger processes, i.e., diffusion processes considered from Schrödinger (1931)’s time-symmetrical point of view. They are uniquely characterized by the large deviation principle deduced in Chapter 5. The associated rate function minimizes the relative entropy with respect to a renormalized Markovian reference process which has in general singular creation and killing as discussed in Sections 6.2 and 6.4. The n-product of the renormalized reference process conditioned by means of the empirical distribution on an approximation of A _a,b in (5.2) possesses a Markovian modification. This system of interacting diffusion processes is proved to perform propagation of chaos in entropy with a Schrödinger process as limiting distribution.