On short time asymptotic behavior of some symmetric diffusions on general state spaces (Q1349060)

The short time behavior of Varadhan's type for heat kernels is considered for a strongly continuous, symmetric Markovian \(L^2\) conservative semigroup on a probability space. In the cases without transition density functions, the lower bound and upper bound of the large deviation rate of transition probability from a set \(A\) to a set \(B\) were given by many authors, such as S. Fang, T. Zhang, S. Aida. In the present paper, the estimation is proven for a general measurable set \(A\) and a set \(B\) in terms of the intrinsic metric. Under some boundedness condition on the associated Dirichlet form, the existence of the corresponding limit for the transition from the set \(A\) to set \(B\) is proven.