Stochastic Analysis: Incompressible Viscous Fluids & SPDEs on Graphs

This lecture delves into the study of incompressible viscous fluids in the plane and stochastic partial differential equations on graphs, focusing on fast advection and non-smooth noise. The instructor, Sandra Cerrai, explores the behavior of the solutions as the small parameter € approaches 0, revealing a non-trivial limit corresponding to an SPDE on a graph. Assumptions on the stream function and random forcing are discussed, leading to the unique mild solution of the equations. The lecture also covers the long-time behavior of the particle density and the convergence of semigroups in weighted spaces. The presentation concludes with a discussion on the well-posedness of the SPDE on the graph and the convergence of solutions.