Schrödinger problem, optimal transport and stochastic calculus – SPOT

The goal of this project is to study the most probable trajectory followed by an interacting particle system when a spontaneous fluctuation is observed. In its most basic form, this is known as the Schrödinger problem. In their general form, the problems at the heart of this project are formulated by means of large deviations theory. Notable examples we shall study include classical models in statistical mechanics and systems of interacting diffusion processes. We have two main objectives. The first is to study the infinite dimensional HJB equations associated to a general Schrödinger problem and to carry out a rigorous analysis of the optimality conditions both in the form of coupled PDE systems and in terms of forward backward pathwise stochastic equations. The second goal is to obtain entropy dissipation estimates along Schrödinger bridges to understand their ergodic behaviour in connection with the obtention of a novel class of functional inequalities and the turnpike property.