Communications inzyxwvutsrqp Commun. Math. Phys. 109, 537 561 (1987)zyxwvutsrqponmlkjihgfedcbaZYXWV Mathematicalzyxwvutsrqp Physics © Springer Verlag 1987 Nonlinear Parabolic Stochastic Differential Equations with Additive Colored Noise onzyxwvutsrqponmlkjihgfedcbaZYXW R d x R + : A Regulated Stochastic Quantization Charles R. Doering* Department of Physics, Center for Relativity and Department of Astronomy, The University of Texas at Austin, Austin, Texas 78712, USA Abstract. We prove the existence of solutions to the nonlinear parabolic stochastic differential equation zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF (d/dt Δ)φ = V'(φ ) +η c for polynomials V of even degree with positive leading coefficient and η c a gaussian colored noise process on R d x R + . When η c iscolored enough that the gaussian solution to the linear problem has Holder 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 area cutoff version of the Parisi Wu stochastic quantization procedure for P{φ ) d models, andthe results of this paper rigorously establish the nonperturbative nature of regularization via modification of the noise process. In the limit η 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. 1. Introduction and Overview This work is motivated bythe stochastic quantization procedure proposed by Parisi and Wu [1]. In this approach, the euclidean field measure for fields on R d with action functional S dμ (φ ) = exp( S[φ ])Π ^M/Jexp( 5[φ ])Π ^W (1.1) is considered as the formal stationary probability distribution of the random process defined by the stochastic differential equation (Langevin equation) * Current address: Center for Nonlinear Studies, MS B258 Los Alamos National Laboratory Los Alamos, NewMexico 87545, USA