Superposition principles for the Zakai equations and the Fokker-Planck equations on measure spaces

TL;DR

This paper establishes superposition principles linking Zakai and Fokker-Planck equations on measure spaces, providing new insights into their solutions and conditions for solvability in weak senses.

Contribution

It proves a superposition principle for Fokker-Planck equations on infinite-dimensional spaces and applies it to Zakai equations, advancing understanding of their solutions.

Findings

Superposition principle for Fokker-Planck equations on $ R^ N$

Superposition principles for Zakai and Fokker-Planck equations on measure spaces

Weak conditions for solving Fokker-Planck equations in the weak sense

Abstract

The work concerns the superposition between the Zakai equations and the Fokker-Planck equations on measure spaces. First, we prove a superposition principle for the Fokker-Planck equations on $\mR^\mN$ under the integrable condition. And then by means of it, we show two superposition principles for the weak solutions of the Zakai equations from the nonlinear filtering problems and the weak solutions of the Fokker-Planck equations on measure spaces. As a by-product, we give some weak conditions under which the Fokker-Planck equations can be solved in the weak sense.