This work is involved with the short-time asymptotics of diffusion semigroups in a general setting. A generalization of Fang's version of Varadhan's formula is proven for general Dirichlet spaces that are local and conservative. The intrinsic metric appearing in the formula is characterized by pointwise distance for canonical Dirichlet spaces on loop groups.
"Small-time Gaussian behavior of symmetric diffusion semi-groups." Ann. Probab. 31 (3) 1254 - 1295, July 2003. https://doi.org/10.1214/aop/1055425779