Evolution systems and continuity equations for white noise functionals

Abstract

In this paper, by applying the quantum decomposition of white noise functionals, the continuity equations for white noise functionals are reformulated as time inhomogeneous white noise differential equations. We establish a systematic study of the evolution systems whose infinitesimal generators are quadratic quantum white noises and then, as an application, we study the continuity equations for white noise functionals of which the coefficient has up to the first chaos (Gaussian process). The restriction for the coefficient is comparable (not necessarily equivalent) to the integrability condition of the exponential of the coefficient by Ambrosio and Figalli [J. Funct. Anal. 256, 179–214 (2009)] for the well-posedness of the continuity equation in the abstract Wiener space setting.