Differentiability of measures associated with parabolic equations on infinite dimensional spaces

The transition measures of the Brownian motion on manifolds modelled on abstract Wiener spaces locally correspond to fundamental solutions of certain infinite dimensional parabolic equations. We establish the existence of such fundamental solutions under a broad new set of hypotheses on the differential coefficients. The fundamental solutions can be approximated in total variation by fundamental solutions of “almost” finite dimensional parabolic equations. By the finite dimensional theory, the approximations are seen to be differentiable. We prove that the property of differentiability is closed under a particular type of sequential convergence, änd conclude the differentiability of the fundamental solutions of the infinite dimensional parabolic equations. This result provides strong evidence in support of the conjecture that the transition measures of the Brownian motion are differentiable, and hence is of importance in the construction of infinite dimensional Laplace-Beltrami operators.