Projet de thèse en Mathematiques
Sous la direction de Marc Arnaudon.
Thèses en préparation à Bordeaux , dans le cadre de DE MATHEMATIQUES ET D'INFORMATIQUE DE BORDEAUX , en partenariat avec IMB Institut de Mathématiques de Bordeaux (laboratoire) depuis le 08-09-2014 .
Defined several decades ago, martingales in manifolds are very canonical objects. They can be defined as infinite interations of conditional barycenters. About these objects very simple questions are still unresolved. For instance, given a random variable with values in a complete manifold and a continuous filtration (one with respect to which all real-valued martingales admit a continuous version), does there exist a continuous martingale in the manifold with terminal value given by this random variable ? Under convex geometry assumptions, this question has received answers in KENDALL-90, PICARD-91, PICARD-94 DARLING-95, ARNAUDON-97. Does there exist one which minimizes the energy, that is, the L2 norm of the quadratic variation ? What about semi-martingales with prescribed drift and terminal value ? Not only are these questions very natural, they are also related to several other interesting problems and useful for many applications. In particular martingales in manifolds allow to define barycenters associated to filtrations, which are sometimes much easier to compute than usual barycenters or means, and which have an associativity property. They are closely related to control theory, stochastic optimization, as well as backward stochastic differential equations (BSDE). Solving by geometric means the problem of existence and uniqueness of martingale with prescribed terminal value yields new solutions to BSDE. If one works on the group of incompressible diffeomorphisms of a compact manifold, such semi-martingales with prescribed terminal value should lead to a solution of the Navier-Stokes equation in a sense to be determined. Equivalent questions in terms of forward backward stochastic differential equations (FBSDE) have been studied in CHEN-CRUZEIRO-QIAN-13, CRUZEIRO-SHAMAROVA-09. Moreover, by changing the energy functional, one expects to find solutions to porous medium equations. See HU-QIAN-ZHANG-12 for the equivalent approach with FBSDE.