Transport equations for a general class of evolution equations with randomperturbations

We derive transport equations from a general class of equations of form iu( t) = H(X,D)u + V(X,D)u where H(X,D) and V(X,D) are pseudodifferential opera tors (Weyl operator) with symbols H(x,k) and V(x,k), where H(x,k) being pol ynomial in k and smooth in x,V(x,k) is a mean zero random function and is s tationary in space variable. We also consider system of equations in the ab ove form. Such equations cover many of the equations that arise in wave pro pagations, such as those considered in a paper by Ryzhik, Papanicolaou, and Keller [Wave Motion 24, 327-370 (1996)]. Our results generalize those by R yzhik, Papanicolau, and Keller. (C) 1999 American Institute of Physics. [S0 022-2488(99)03209-0].