Short-time asymptotics in Dirichlet spaces (Q2710681)

In 1967 Varadhan proved that the heat kernel \(p_t(x,y)\) on a Riemannian manifold has the following asymptotic behavior as \(t\) gets small: \(\lim_{t\to 0}t\log p_t(x,y)= -\tfrac 12 d^2_a(x,y),\) where \(d_a(x,y):=\sup_{\langle df/dx,a df/dx\rangle\leq 1} f(x)-f(y)\). Since then, the question of whether it is possible to recover Varadhan's formula in a more general setting has been investigated in many papers. Some generalizations for general operators in \(R^n\) and to Lipschitz manifolds have been made. Then the question came of what happens in infinite dimensions. The work started with Fang (1994), who proved a version of this result for the Ornstein-Uhlenbeck process on Wiener space. The present work studies the short-time asymptotics of the heat semigroup in a general setting. The main result is to prove a generalization of Fang's version of Varadhan's formula.