Ben Fehrman - Non-equilibrium fluctuations in interacting particle systems and conservative stochastic PDE

CDT Workshops

4 November 16:15

Mathematical Institute, University of Oxford

Ben Fehrman

Ben Fehrman - Non-equilibrium fluctuations in interacting particle systems and conservative stochastic PDE

The Dean-Kawasaki equation, and more generally certain singular stochastic PDEs with conservative space-time white noise, arise formally in fluctuating hydrodynamics and macroscopic fluctuation theory to describe far from equilibrium behavior in physical systems, such as the fluctuations of an interacting particle system about its hydrodynamic limit. The treatment of these SPDEs presents a significant mathematical challenge, due both to their supercriticality and their degenerate and singular coefficients.

In this talk, which is based on joint work with Benjamin Gess, I will discuss a well-posedness theory for such equations with correlated noise. The introduction of smooth noise is justified by the fact that discrete microscopic systems often have a natural correlation scale, such as the grid-size, and by the fact that, along appropriate scaling limits, we prove that the solutions accurately describe the particle system in terms of a law of large numbers, central limit theorem, and large deviations principle. The methods treat general nonlinearities that are only locally 1/2-Hölder continuous, and solve several open problems including the well-posedness of the Dean-Kawasaki and nonlinear Dawson-Watanabe equations with correlated noise.