Speaker
Description
The superposition principle turns a solution of a (possibly nonlinear) Fokker–Planck equation into a probability measure on path space that solves the associated martingale problem. It does not, in general, produce a Markov process, and the strong Markov property for such path laws has remained open in general even in the linear case. In the first part of the talk we show how potential theory closes this gap. Under merely measurable conditions on the coefficients, we construct a right (hence strong Markov) process on a large set of space-time starting points whose one-dimensional marginals reproduce the given solution. For McKean–Vlasov SDEs with Nemytskii-type coefficients, such as those behind generalized porous media equations, this gives the strong Markov property for the corresponding nonlinear Markov process. In the second part we use these processes as the motion of a branching particle system. Picture a large population of interacting healthy particles, described in the mean-field limit by a McKean–Vlasov SDE, and a small population of infected particles that move like the healthy ones and branch at rates depending on time, position and the distribution of the whole population. We construct the infected population as a right branching process on the space of finite configurations. The approach is analytic: the branching semigroup is a jump perturbation of the diagonal process, realized through inverse subordination. This avoids the regularity conditions on the base process required by earlier constructions, and it identifies the fine and natural topologies of the resulting process. Joint work with Lucian Beznea and Michael Röckner.