The Parabolic Harnack Inequality for the Time Dependent Ginzburg–Landau Type SPDE and its Application

Abstract

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.