Nonlinear parabolic stochastic differential equations with additive colored noise onR ^d ×R ₊: A regulated stochastic quantization

Abstract

We prove the existence of solutions to the nonlinear parabolic stochastic differential equation

$$({\partial \mathord{\left/ {\vphantom {\partial {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}} - \Delta )\varphi = - V'(\varphi ) + \eta _c $$

for polynomialsV of even degree with positive leading coefficient and ν _c a gaussian colored noise process onR ^d×R ₊. When ν _c is colored enough that the gaussian solution to the linear problem has Hölder continuous covariance, the nongaussian processes are almost surely realized by continuous functions. Uniqueness, regularity properties, asymptotic perturbation expansions and nonperturbative fluctuation bounds are obtained for the infinite volume processes. These equations are a cutoff version of the Parisi-Wu stochastic quantization procedure forP(ϕ) _d models, and the results of this paper rigorously establish the nonperturbative nature of regularization via modification of the noise process. In the limit ν _c → gaussian white noise we find that the asymptotic expansion and the rigorous bounds agree for processes corresponding to the (regulated) stochastic quantization of super-renormalizable and small coupling, strictly renormalizable scalar field theories and disagree for nonrenormalizable models.