Smooth solutions of quasilinear stochastic partial differential equations of McKean-Vlasov type (Q2726265)

Smooth solutions of quasilinear stochastic partial differential equations (SPDE) of McKean-Vlasov type are studied. The position of the \(N\)-particle system at time \(t\) as a point in \(R^{dN}\) is considered and a description of its time evolution is a microscopic model for the particle distribution. Typically, ordinary differential equations or stochastic differential equations for the position are derived from mechanical principles. Such equations are called microscopic equations. The empirical mass distribution at time \(t\) of the \(N\)-particle system (with the mass \(a_{i}\) of the \(i\)th particle) is given by linear combination of the point measures, concentrated in \(r\in R^{d}.\) It is a measure process. A mass distribution is called a macroscopic one, if the particle system cannot be seen and stochastic fluctuations are absent. If stochastic fluctuations are present, it is called a mezoscopic model. The author derived [Probab. Theory Relat. Fields 102, No. 2, 159-188 (1995; Zbl 0821.60066)] a SPDE for the above mentioned measure process which was called a mezoscopic equation. A particle approach to derive mezoscopic equations was first used by \textit{D. A. Dawson} [J. Multivariate Anal. 5, 1-52 (1975; Zbl 0299.60050)] for a system of branching and diffusing particles. The author's approach was first developed for a model of interacting and diffusing vortices in a 2D-fluid. Microscopic equations for \(N\)-particles with singular interaction were analyzed by Skorokhod in 1996.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].