The Gaussian effective potential and stochastic partial differential equations

Abstract

We investigate arbitrary stochastic partial differential equations subject to translation invariant and temporally white noise correlations from a nonperturbative framework. The method that we expose first casts the stochastic equations into a functional integral form, then it makes use of the Gaussian effective potential approach, which is an useful tool for describing symmetry breaking. We apply this method to the Kardar–Parisi–Zhang equation and find that the system exhibits spontaneous symmetry breaking in ( 1 + 1 ) , ( 2 + 1 ) and ( 3 + 1 ) Euclidean dimensions, providing insight into the evolution of the system configuration due to the presence of noise correlations. A simple and systematic approach to the renormalization, without explicit regularization, is employed.