Covariance structure of parabolic stochastic partial differential   equations
  In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space-time weak variational formulation of this tensorized equation is established. 
Annika Lang, Stig Larsson and Christoph Schwab
math.PR