THE STOCHASTIC BURGERS EQUATION: FINITE MOMENTS AND SMOOTHNESS OF THE DENSITY

Abstract

We apply a stochastic version of the Hopf–Cole transformation to the solution of the stochastic Burgers equation with Dirichlet boundary conditions driven by a space–time Gaussian white noise. As a consequence, we deduce that the solution of the Burgers equation has moments of all orders. Using the techniques of the Malliavin calculus we show that if the dispersion is state-independent, then the solution of the stochastic Burgers equation has a smooth density at any point (t, x), with t>0 and x ∈(0,1).