The parabolic Harnack inequality for the time dependent Ginzburg-Landau type SPDE and its application

The main purpose of this paper is to establish the parabolic Harnack inequality for the transition semigroup associated with the time dependent Ginzburg-Landau type stochastic partial differential equation (=SPDE, in abbreviation). In view of quantum field theory, this dynamics is called a P(φ)₁-time evolution. We prove the main result by adopting a stochastic approach which is different from Bakry-Emery's Γ₂- method. As an application of our result, we study some estimates on the transition probability for our dynamics. We also discuss the Varadhan type asymptotics.