Decoupled mild solutions of deterministic evolution problemswith singular or path-dependent coefficients, represented by backward SDEs.

Summary This thesis introduces a new notion of solution for deterministic nonlinear evolution equations, called mild decoupled solutions. We revisit the links between Brownian Markovian backward differential equations (BMEEs) and semilinear parabolic PDEs by showing that, under very weak assumptions, BMEEs produce a unique mild decoupled solution of a PDE.We extend this result to many other deterministic equations such as Pseudo-EDPs, Integral Partial Derivative Equations (IPDEs), distributional drift PDEs, or trajectory dependent E(I)DPs. The solutions of these equations are represented via EDSRs which can be without reference martingale, or directed by cadlag martingales. In particular, this thesis solves the identification problem, which consists, in the classical case of a Markovian EDSR, in giving an analytical meaning to the process Z, second member of the solution (Y,Z) of the EDSR. In the literature, Y usually determines a viscosity solution of the deterministic equation and this identification problem is solved only when this viscosity solution has a minimum of regularity. Our method allows to solve this problem even in the general case of jumping EDSRs (not necessarily Markovian).