The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities

We study the mean field Schrödinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud of interacting Brownian particles conditionally on the observation of their initial and final configuration. Its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent Brownian particles. We undertake a fine study of the dynamics of its solutions, including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the time between observations grows larger and larger, as well as a novel class of functional inequalities involving the mean field entropic cost (i.e. the optimal value in (MFSP)). Our strategy unveils an interesting connection between forward backward stochastic differential equations and the Riemannian calculus on the space of probability measures introduced by Otto, which is of independent interest.

Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains

We consider the maximum entropy Markov chain inference approach to characterize the collective statistics of neuronal spike trains, focusing on the statistical properties of the inferred model. We review large deviations techniques useful in this context to describe properties of accuracy and convergence in terms of sampling size. We use these results to study the statistical fluctuation of correlations, distinguishability and irreversibility of maximum entropy Markov chains. We illustrate these applications using simple examples where the large deviation rate function is explicitly obtained for maximum entropy models of relevance in this field.

ANALYTICAL MECHANICS IN STOCHASTIC DYNAMICS: MOST PROBABLE PATH, LARGE-DEVIATION RATE FUNCTION AND HAMILTON–JACOBI EQUATION

Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Lagrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation dXt = b(Xt)dt+ϵdWt, where Wt is a Brownian motion. In the limit of vanishingly small ϵ, the solution to the stochastic differential equation other than [Formula: see text] are all rare events. However, conditioned on an occurrence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with [Formula: see text] and Hamiltonian equations with H(p, q) = ‖p‖2+b(q)⋅p. Hamiltonian conservation law implies that the most probable trajectory for a "rare" event has a uniform "excess kinetic energy" along its path. Rare events can also be characterized by the principle of large deviations which expresses the probability density function for Xt as f(x, t) = e-u(x, t)/ϵ, where u(x, t) is called a large-deviation rate function which satisfies the corresponding Hamilton–Jacobi equation. An irreversible diffusion process with ∇×b≠0 corresponds to a Newtonian system with a Lorentz force [Formula: see text]. The connection between stochastic motion and analytical mechanics can be explored in terms of various techniques of applied mathematics, for example, singular perturbations, viscosity solutions and integrable systems.