On nonlinear Markov Processes in the sense of McKean – Marco Rehmeier (Bielefeld University)


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We study nonlinear Markov processes in the sense of McKean and present a large new class of examples. Our notion of nonlinear Markov property is in McKean’s spirit, but more general in order to include examples of such processes whose one-dimensional time marginals solve a nonlinear parabolic PDE, such as Burgers’ equation, the porous media equation, or variants of the latter with transport-type drift. We show that the associated nonlinear Markov process is given by path laws of weak solutions to a corresponding distribution-dependent stochastic differential equation whose coefficients depend singularly (i.e. Nemytskii-type) on its one-dimensional time marginals. Moreover, we show that also for general nonlinear Markov processes, their path laws are uniquely determined by one-dimensional time marginals of suitable associated conditional path laws. Furthermore, we characterize the extremality of the curves of the one-dimensional time marginals of our nonlinear Markov Processes in the class of all solutions to the associated linearized PDE and, this way, obtain new interesting results also for the classical linear case. This is joint work with Michael Röckner.

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