An integrated version of Varadhan's asymptotics for lower-order   perturbations of strong local Dirichlet forms
  The studies of Ram\'irez, Hino-Ram\'irez, and Ariyoshi-Hino showed that an integrated version of Varadhan's asymptotics holds for Markovian semigroups associated with arbitrary strong local symmetric Dirichlet forms. In this paper, we consider non-symmetric bilinear forms that are the sum of strong local symmetric Dirichlet forms and lower-order perturbed terms. We give sufficient conditions for the associated semigroups to have asymptotics of the same type. 
Masanori Hino and Kouhei Matsuura
math.PR math.AP