Large Deviations

Abstract

Schrödinger processes in Definition 2.4.1 are diffusion processes with given dynamics and prescribed marginal distributions at finite initial time a and final time b. Section 1.3 provides a large deviation approach to Schrödinger processes in a discrete setting. This chapter is devoted to a large deviation principle for the set A _a,b of probability measures on a path space which have the prescribed pair (μ_aμ_b) of probability measures as initial and final distribution, respectively. The formulation of a so-called approximate Sanov property of A _a,b in Section 5.1 is not straightforward. In fact, its interior Å _a,b depending on the topology can be empty and empirical distributions taking only discrete values do not belong to the set A _a,b in general. Hence an approximation of A _a,b by enlarged sets of probability measures is going to be analyzed by means of Csiszar’s τ₀-topology.