Stochastic Burgers Equation with Random Initial Velocities: A Malliavin Calculus Approach

Abstract. In this article we prove an existence theorem for solutions of the stochastic Burgers equation (SBE) on the unit interval with Dirichlet boundary conditions and anticipating initial velocities. The SBE is driven by affine (additive + linear) noise. In order to establish the existence theorem, we adopt a somewhat counterintuitive perspective in which stochastic dynamical systems ideas lead to the existence of solutions rather than vice versa. More specifically, our approach uses the Malliavin calculus and is based on the existence and regularity of a perfect cocycle on the energy space for the SBE. The proof of the existence theorem requires Malliavin regularity of the infinitedimensional initial velocity field together with new spatial estimates on the cocycle, its Fréchet and Malliavin derivatives. The existence theorem provides a dynamic characterization of solutions of the nonanticipating SBE on its unstable invariant manifolds. Furthermore, as a corollary of the existence theorem, we show that random cocycle-invariant points on the energy space correspond to (possibly nonergodic) stationary pathwise solutions for the SBE.