Lagrangian stochastic models of conditional McKean-Vlasov type and their containment.

Summary In this thesis, we are interested in the theoretical aspects of a new class of stochastic differential equations called Lagrangian stochastic models. These models have been introduced to model the properties of particles associated with turbulent flows. Motivated by a recent application of these models in the development of scale refinement methods for weather forecasting, we also consider the introduction of edge conditions in the dynamics. In the framework of McKean-type nonlinear equations, Lagrangian stochastic models designate a particular class of nonlinear dynamics due to the presence in the coefficients of conditional distribution. In simplified cases, we establish the well-posedness of these dynamics and their particle approximation. Concerning the introduction of edge conditions, we construct a stochastic confined model for the prototype condition of "no permeability on average". In the case where the confining domain is the hyperplane, we obtain an existence and uniqueness result for the considered dynamics, and show that the edge condition is satisfied. For general domains, we study the conditional McKean-Vlasov-Fokker-Planck equation satisfied by the law of systems. We develop the notions of Maxwellian over- and under-solutions, giving the existence of Gaussian bounds on the solution of the equation.