| Abstract: |
| For a solution to a nonlinear Fokker-Planck equation (FPE) the powerful superposition principle renders a probability measure on path space with one dimensional time marginals equal to this solution, and additionally solving the martingale problem for the Kolmogorov operator given by the FPE. The superposition principle thus reveals that such parabolic PDEs have a probabilistic counter part. The aim of this talk is to go a substantial further step and, by exploiting the superposition principle, Dirichlet forms, and potential theoretic tools, construct a full fledged Markov process, i.e. a family of path space measures for a large set of space time starting points connected by the Markov property, associated to the (linearized) FPE in the above way. Under very general (merely measurability) conditions on the coefficients of the FPE this is achieved in this paper in such a way that the resulting process is a right process. Furthermore, we introduce a Choquet capacity for such FPEs using the corresponding right process. A main application here is the FPE given by the generalized porous media equation and its corresponding McKean-Vlasov SDE. |
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