Axis 7: PDE: stochastic approches

This research axis consists in studying the properties of some classes of PDEs (existence, uniqueness, long time behavior, regularity …) using stochastic processes. The study of systems of progressive retrograde stochastic differential equations (SDE-RSDEs) allows for example to obtain a probabilistic representation for such PDEs, representation commonly called Feynman-Kac formula. This representation also helps to build and to study the convergence of probabilistic algorithms to numerically solve these PDEs.

The SDE-RSDEs also allow us to model the equations of hydrodynamics and their approximate solution give new simulation methods. Their combination with variational methods to answer questions of existence of generalized flux with initial and final conditions. RSDEs are also a promising tool for smoothing and denoising signals via the design of martingales with given terminal value.