On the Stochastic Burgers’ Equation in the Real Line

Abstract

In this paper we establish the existence and uniqueness of an $L^2(\mathbb{R})$ -valued solution for a one-dimensional Burgers’ equation perturbed by a space–time white noise on the real line. We show that the solution is continuous in space and time, provided the initial condition is continuous. The main ingredients of the proof are maximal inequalities for the stochastic convolution, and some a priori estimates for a class of deterministic parabolic equations.