Geometric Analysis for Symmetric Fleming–Viot Operators: Rademacher's Theorem and Exponential Families

Abstract

We use the natural geometry of a symmetric Fleming–Viot operator ℒ to obtain analytical descriptions of the corresponding Dirichlet space (E,D(E)). In particular, we give a complete characterization of functions in D(E) in terms of their differentiability properties along exponential families. Moreover, we prove a Rademacher theorem stating that any function which is Lipschitz continuous with respect to the Bhattacharya distance is contained in D(E) and possesses a bounded gradient. A converse to this statement is also given. Thus, we relate the Bhattacharya distance to the potential theory of ℒ.