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. [References: 30]
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