Bismut Formulae and Applications for Functional SPDEs
  By using Malliavin calculus, explicit derivative formulae are established for a class of semi-linear functional stochastic partial differential equations with additive or multiplicative noise. As applications, gradient estimates and Harnack inequalities are derived for the semigroup of the associated segment process. Keywords: Bismut formula, Malliavin calculus, gradient estimate, Harnack inequality, functional SPDE 
Jianhai Bao, Feng-Yu Wang, Chenggui Yuan
math.PR