Transport equations for a general class of evolution equations with random perturbations

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 operators (Weyl operator) with symbols H(x,k) and V(x,k), where H(x,k) being polynomial in k and smooth in x,V(x,k) is a mean zero random function and is stationary in space variable. We also consider system of equations in the above form. Such equations cover many of the equations that arise in wave propagations, such as those considered in a paper by Ryzhik, Papanicolaou, and Keller [Wave Motion 24, 327-370 (1996)]. Our results generalize those by Ryzhik, Papanicolau, and Keller. (C) 1999 American Institute of Physics. [S0022-2488(99)03209-0].Physics, MathematicalSCI(E)11ARTICLE104828-48584