We are interested in the propagation of a Gaussian variable through a nonlinear stochastic differential equation (SDE). This problem has applications in Kalman filtering where the SDE models the dynamic of a system under uncertainties but also in density estimation where the SDE is a Langevin dynamic which converges in distribution to a Gibbs density. To do this, we approximate the solution of the Fokker-Planck equation with a Gaussian considering a Bures-JKO scheme and show the obtained Gaussian flow converges exponentially fast to the Gibbs target if it's log-concave. Otherwise, we lose global convergence guarantees but we can still compute a flow of Gaussian mixture which approximates it empirically very well. This presentation will be based on the following paper: https://arxiv.org/abs/2205.15902.
Camille Labourie