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A random wave process. (English) Zbl 0564.60061
The description of waves in a random medium propagating mainly in one direction leads to the random Schrödinger equation \(i \partial V/\partial t+\Delta V+\mu V=0,\) \(x\in (x_ 1,x_ 2)\in {\mathbb{R}}^ 2,\) \(V(0,x_ 1,x_ 2)=V_ 0(x_ 1,x_ 2),\) \(\Delta =\partial^ 2/\partial x^ 2_ 1+\partial^ 2/\partial x^ 2_ 2,\) \(\mu (t,x_ 1,x_ 2)\) a given real-valued process. Under the assumption that \(\mu\) is Gaussian white noise in t a unique mild solution of the corresponding Fourier-transformed stochastic partial differential equation is constructed in \(L^ 2({\mathbb{R}}^ 2)\) by a Wiener-Itô expansion. This solution turns out to be a Markov diffusion process on the unit sphere of \(L^ 2({\mathbb{R}}^ 2)\) (conservation of energy). A limiting regime (narrow beam spot-dancing) is investigated.
Reviewer: L.Arnold

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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