SPOT is a research project funded by the French ANR (National Research Agency) over the years 2021-2024.
Project description
We focus on the general problem of studying the most likely evolution of an interacting particle system conditionally on the observation of a macroscopic fluctuation. This problem, called the Schroedinger problem has been introduced by E. Schroedinger himself in 1931 when the underlying particle system is made of independent Brownian particles.
The goal of this project is to undertake a systematic study of instances of the Schrodinger problem for particle systems whose interaction mechanism is of mean field type. In all these cases, the problem of finding the most likely evolution is cast as the problem of minimising the large deviations rate function of the empirical (path) measure of the underlying particle system. Important examples include lattice gases (e.g. simple exclusion, zero range) and weakly interacting diffusions (e.g. Mc Kean Vlasov dynamics) and the Langevin dynamics. Among our objectives are
The dynamic characterisation of optimisers, called Schroedinger bridges, via forward-backward stochastic differential equations, coupled PDE systems, or measure valued Hamilton Jacobi Bellman equations. We are also interested in the geometric interpretation of these equations by means of an appropriate Riemannian formalism. In the last year the Riemannian structure associated with optimal transport, called Otto calculus, has proven to be extremely useful in the analysis of the classical SP. In this project, we aim at exploring other Riemannian structures, in particular those associated with non linear mobilities in the context of the Schroedinger problem for lattice gases.
The study of the long time behaviour Schroedinger bridges, in connection with the turnpike phenomenon in stochastic optimal control and the obtention of novel classes of functional inequalities. In particular, by using the Schrodinger as an entry point, we aim at developing a robust tools for estimating the exponential convergence of dynamic stochastic control problems towards their ergodic limit.
Project Members
Giovanni Conforti (École Polytechnique)
Laurent Pfeiffer (Centrale Supélec)
Zhenjie Ren (Université Paris Dauphine)
Luca Tamanini (Università Bocconi di Milano)
Daniela Tonon (Università di Padova)