Stochastic and deterministic analysis for irregular models

Abstract

The ambition of the project consists in describing and investigate irregular phenomena arising from hydrodynamics, oncology, economics, or complex systems, from a macroscopic-microscopic point of view. We will take advantage of the complementarity of deterministic and stochastic analysis. Many difficulties appear such as discontinuous (even distributional) coefficients, jumps, rough behaviour of stochastic processes, non-conservativity, and lack of Markovian character. Typical examples corresponding to real applications areKeller-Segel models, Burgers-Huxley, fast and superfast diffusions, porous media type equations, self-organized criticality, McKean SDEs in random environment, Vlasov-Navier-Stokes, Hamilton-Jacobi PDEs. We are guided by three main motivations.1) Deterministic macroscopic modeling and mathematical theory.2) Stochastic microscopic modeling involving extended McKean probabilistic representations of irregular models together with particle approximations.3) Numerical simulation: to provide approximation schemes for non-linear PDEs potentially involving path-dependent coefficients, and random environment. (AU)