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

Abstract

We derive transport equations from a general class of equations of form iut=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.

Transport equations for waves in randomly perturbed Hamiltonian systems, with application to Rossby waves

“…The Hamiltonian nature of the systems considered results in scattering terms that are clearly conservative, unlike those of Guo & Wang [3] who treated general, non-Hamiltonian Schrödinger equations.…”

“…They generalise the results of Ryzhik et al [10] and of Guo & Wang [3] (in the conservative case) to the large class of randomly perturbed linear Hamiltonian systems of the form (1.2). With these results, the derivation of transport equations for particular systems is reduced to the straightforward, algorithmic computation of the left and right eigenvectorŝ e (s) (x, t) and e (s) (x, t) and of their products with the correlation 4-tensors defined in (2.6)-(2.8).…”

“…We have chosen this interpretation of pseudodifferential operators rather than the Weyl correspondence chosen by Guo & Wang [3] for its simplicity, even though it makes our derivation of the transport equation somewhat less elegant; it is straightforward, if sometimes cumbersome, to translate between the two interpretations [see, e.g., 2]. With the assumption that J(x, ∂¡ ) and H(x, ∂¡ ) are, respectively, skew-adjoint and self-adjoint, the conservation of the wave energy (1.3) is readily established.…”

Radiative transport limit for the random Schrödinger equation

“…Moreover, the test functions that we used in our proof of Theorem 2.1 are based on the formal expressions for W (1) and W (2) . The role of the regularization parameter δ is played by the spectral gap of the generator Q because of the bound (14).…”

“…Formal derivations of radiative transport equations for various types of waves in random media are given in [2,14,22]. Appendix A contains such a derivation for the time-dependent case.…”