Statistical solutions of PDEs by nonstandard densities (Q803437)

Statistical solutions of partial differential equations are discussed in the framework of nonstandard analysis. Statistical solution of a PDE is a family of measures on a Hilbert space of functions, which satisfies a certain integral equation. Using nonstandard analysis infinite dimensional spaces can be represented by hyperfinite (formally finite) dimensional ones which allows us to describe the measures on function spaces by means of densities with respect to standard version of Lebesgue measure. The equation characterizing the measures can then be replaced by a linear first order PDE for their densities which is solvable by the method of characteristics. This provides a method of constructing statistical solutions. As a tool a hyperfinite approach to weak solutions of PDEs is presented.