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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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##### References:
 [1] Dawson DA and Papanicolaou GC (1984) Waves in random media in the forward scattering approximation (in press) [2] Dawson DA and Kurtz TG (1982) Application of duality to measure-valued processes. In: Fleming W and Gorostiza LG (eds) Lecture Notes in Control and Information Science, vol. 42:91-105. Springer-Verlag, New York [3] Dawson DA and Salehi H (1982) Spatially homogeneous random evolutions. J Mult Anal 10:141-180 · Zbl 0439.60051 [4] Furutsu K (1982) Statistical theory of wave propagation in a random medium and the irradiance distribution function. J Opt Soc Amer 62:240-254 [5] Furutsu K and Furuhama Y (1973) Spot dancing and relative saturation phenomena of irradiance scintillation of optical beams in a random medium. Optica 20:707-719 [6] Holley R and Stroock D Generalized Ornstein-Uhlenbeck Processes as limits of interacting systems, In: Williams D (ed) ?Stochastic Integrals?, Lecture Notes in Mathematics, vol. 851:152-168. Springer-Verlag, New York [7] Itô K (1983) Stochastic differential equations in infinite dimensions. CBMS-NSF Regional Conference. SIAM, Philadelphia [8] Keller JB (1964) Stochastic equations and wave propagation in random media. Proc Symp Appl Math 16:145-170 [9] Klyatskin VI and Tatarskii VI (1970) A new method of successive approximations in the problem of the propagation of waves in a random medium having random large-scale inhomogenieties. Radiophys and Quantum Electronics (USSR) 14:1110-1111 [10] Klyatskin VI (1975) Statistical description of dynamical systems with fluctuating parameters (in Russian). Nauka, Moscow [11] Miyahara Y (1982) Stochastic evolution equations and white noise analysis. Ottawa: Carleton Mathematical Lecture Notes No. 42 · Zbl 0489.60070 [12] Meiden R (1980) On the connection between ordinary and generalized stochastic processes. J Math Anal Appl 76:124-133 · Zbl 0453.60040 [13] Stratonovich RL (1965) Conditional Markov Processes. Elsevier: New York [14] Strohbehn JW (1978) Laser beam propagation in the atmosphere. Springer-Verlag: New York [15] Stroock DW and Varadhan SRS (1979) Multidimensional diffusion processes. Springer-Verlag: Berlin, Heidelberg, New York [16] Tatarskii VI (1971) The effects of the turbulent atmosphere on wave propagation. National Technical Service: Springfield, VA [17] Trotter HF (1958) Approximation of semigroups of operators. Pac J Math 8:887-919 · Zbl 0099.10302
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