The project will develop stochastic and numerical methods for the study of boundary effects in nonlinear partial differential equations with physical relevance. We will use infinite dimensional stochastic analysis tools, in particular, measure-valued branching Markov processes.
We will provide probabilistic representations for solutions of Navier-Stokes equations in bounded planar domains through an appropriate nonlocal branching process and prove the convergence of the corresponding numerical algorithm. In the case of less regular domains, such as Lipschitz or polygonal domains we will provide an intuitive understanding of how the loss of regularity is related to the geometry of the domain through the branching process. Moreover, we will provide a probabilistic representation for the solutions of nonlinear Neumann boundary problems and corresponding numerical algorithms. Using stochastic differential equations we will derive probabilistic interpretations fragmentation and coagulation processes in duality. These will be applied to the study of avalanche models.