Regularised variational schemes for non-gradient systems, and large deviations for a class of reflected McKean-Vlasov SDE

This thesis consists of two parts. The first part constructs entropy regularised variational schemes for a range of evolutionary partial differential equations (PDEs), not necessarily in gradient flow form, with a focus on kinetic models. The second part obtains Freidlin-Wentzell large deviation principles and exit times for a class of reflected McKean-Vlasov stochastic differential equations (SDEs). The theory ofWasserstein gradient flows in the space of probability measures has made enormous progress over the last twenty years. It constitutes a unified and powerful framework in the study of dissipative PDEs, providing the means to prove well-posedness, regularity, stability and quantitative convergence to the equilibrium. The recently developed entropic regularisation technique paves the way for fast and efficient numerical methods for solving these gradient flows. However, many PDEs of interest do not have a gradient flow structure and, a priori, the theory is not applicable. In the first part of the thesis, we develop time-discrete entropy regularised, (one-step and two-step), variational schemes for general classes of non-gradient PDEs. The convergence of the schemes is proved as the time-step and regularisation strength tend to zero. For each scheme we illustrate the breadth of the proposed framework with concrete examples. In the second part of the thesis we study reflected McKean-Vlasov diffusions over a convex, non-bounded domain with self-stabilizing coefficients that do not satisfy the classical Wasserstein Lipschitz condition. For this class of problems we establish existence and uniqueness results and address the propagation of chaos. Our results are of wider interest: without the McKean-Vlasov component they extend reflected SDE theory, and without the reflective term they extend the McKean-Vlasov theory. Using classical tools from the theory of Large Deviations, we prove a Freidlin-Wentzell type Large Deviation Principle for this class of problems. Lastly, under some additional assumptions on the coefficients, we obtain an Eyring-Kramer's law for the exit time from subdomains contained in the interior of the reflecting domain. Our characterization of the rate function for the exit-time distribution is explicit.