Kinetic limits for waves in a random medium

Guillaume Bal; Tomasz Komorowski; Lenya Ryzhik

doi:10.3934/krm.2010.3.529

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December 2010, 3(4): 529-644. doi: 10.3934/krm.2010.3.529

Kinetic limits for waves in a random medium

Guillaume Bal ^1, , Tomasz Komorowski ^2, and Lenya Ryzhik ^3,

1.	Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027
2.	Institute of Mathematics, UMCS, pl. Marii Curie-Skłodowskiej 1, 20-031, Lubin, IMPAN, ul. Śniadeckich 8, 00-956 Warsaw, Poland
3.	Department of Mathematics, Stanford University, Stanford, CA 94305, United States

Received September 2010 Revised October 2010 Published October 2010

Abstract
Related Papers

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Keywords: Waves in random media, Kinetic equation, Scintillation function, Time reversal., Wigner transform, Wave-wave correlation, Radiative transport equation, Self-averaging, Particles in random flow, Fokker-Planck equation.

Mathematics Subject Classification: Primary: 60H25; Secondary: 35Q4.

Citation: Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic and Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529

References:

[1]	F. Bailly, J. F. Clouet and J.-P. Fouque, Parabolic and gaussian white noise approximation for wave propagation in random media, SIAM J. Appl. Math, 56 (1996), 1445-1470.
[2]	G. Bal, On the self-averaging of wave energy in random media, Multiscale Model. Simul., 2 (2004), 398-420.
[3]	G. Bal, Kinetics of scalar wave fields in random media, Wave Motion, 43 (2005), 132-157.
[4]	G. Bal, Inverse problems in random media: A kinetic approach, J. Phys. Conf. Series, 124 (2008), 012001.
[5]	G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001.
[6]	G. Bal, L. Carin, D. Liu and K. Ren, Experimental validation of a transport-based imaging method in highly scattering environments, Inverse Problems, 26 (2007), 2527-2539.
[7]	G. Bal, D. Liu, S. Vasudevan, J. Krolik and L. Carin, Electromagnetic time-reversal imaging in changing media: Experiment and analysis, IEEE Trans. Anten. and Prop., 55 (2007), 344-354.
[8]	G. Bal, T. Komorowski and L. Ryzhik, Self-averaging of Wigner transforms in random media, Comm. Math. Phys., 242 (2003), 81-135.
[9]	G. Bal, T. Komorowski and L. Ryzhik, Asymptotics of the solutions of the random Schródinger equation, to appear in Arch. Rat. Mech., (2010).
[10]	G. Bal, G. Papanicolaou and L. Ryzhik, Radiative transport limit for the random Schrödinger equations, Nonlinearity, 15 (2002), 513-529.
[11]	G. Bal, G. Papanicolaou and L. Ryzhik, Self-averaging in time reversal for the parabolic wave equation, Stochastics and Dynamics, 4 (2002), 507-531.
[12]	G. Bal and O. Pinaud, Time reversal-based imaging in random media, Inverse Problems, 21 (2005), 1593-1620.
[13]	G. Bal and O. Pinaud, Accuracy of transport models for waves in random media, Wave Motion, 43 (2006), 561-578.
[14]	G. Bal and O. Pinaud, Kinetic models for imaging in random media, Multiscale Model. Simul., 6 (2007), 792-819.
[15]	G. Bal and O. Pinaud, Self-averaging of kinetic models for waves in random media, Kinetic Related Models, 1 (2008), 85-100.
[16]	G. Bal and O. Pinaud, Imaging using transport models for wave-wave correlations, to appear in M3AS, (2011).
[17]	G. Bal and K. Ren, Transport-based imaging in random media, SIAM J. Applied Math., 68 (2008), 1738-1762.
[18]	G. Bal and L. Ryzhik, Time reversal and refocusing in random media, SIAM J. Appl. Math., 63 (2003), 1475-1498.
[19]	G. Bal and L. Ryzhik, Time splitting for wave equations in random media, preprint, (2004).
[20]	G. Bal and L. Ryzhik, Stability of time reversed waves in changing media, Discrete Contin. Dyn. Syst., 12 (2005), 793-815.
[21]	G. Bal and R. Verástegui, Time reversal in changing environment, Multiscale Model. Simul., 2 (2004), 639-661.
[22]	W. Bao, S. Jin and P. A. Markowich, On Time-Splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comp. Phys., 175 (2002), 487-524.
[23]	P. Billingsley, "Convergence of Probability Measures," Wiley, 1999.
[24]	P. Blankenship and G. C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbances. i, SIAM J. Appl. Math., 34 (1978), 437-476.
[25]	P. Blomgren, G. C. Papanicolaou and H. Zhao, Super-Resolution in Time-Reversal Acoustics, J. Acoust. Soc. Am., 111 (2002), 230-248.
[26]	B. Borcea, G. C. Papanicolaou and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems, 19 (2003), S139-S164.
[27]	B. Borcea, G. C. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460.
[28]	B. Borcea, G. C. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems, 22 (2006), 1405-1436.
[29]	A. N. Borodin, A limit theorem for solutions of differential equations with random right hand side, Teor. Veroyatn. Ee Primen, 22 (1977), 498-512.
[30]	R. A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc, 108 (1994), viii+125.
[31]	S. Chandrasekhar, "Radiative Transfer," Dover Publications, New York, 1960.
[32]	G. C. Cohen, "Higher-Order Numerical Methods for Transient Wave Equations," Scientific Computation, Springer Verlag, Berlin, 2002.
[33]	D. A. Dawson and G. C. Papanicolaou, A random wave process, Appl. Math. Optim., 12 (1984), 97-114.
[34]	D. Dürr, S. Goldstein and J. Lebowitz, Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model, Comm. Math. Phys., 113 (1987), 209-230.
[35]	D. R. Durran, "Numerical Methods for Wave equations in Geophysical Fluid Dynamics," Springer, New York, 1999.
[36]	G. F. Edelmann, T. Akal, W. S. Hodgkiss, S. Kim, W. A. Kuperman and H. C. Song, An initial demonstration of underwater acoustic communication using time reversal, IEEE J. Oceanic Eng., 27 (2002), 602-609.
[37]	L. Erdös and H. T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Comm. Pure Appl. Math., 53 (2000), 667-735.
[38]	L. Evans and M. Zworski, Lectures on semiclassical analysis,, Berkeley., ().
[39]	A. Fannjiang, Self-averaging in scaling limits for random high-frequency parabolic waves, Archives of Rational Mechanics and Analysis, 175 (2005), 343-387.
[40]	J. P. Fouque, La convergence en loi pour les processus à valeur dans un espace nucléaire, Ann. Inst. H. Poincaré Prob. Stat, 20 (1984), 225-245.
[41]	J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media," Springer Verlag, New York, 2007.
[42]	P. Gérard, Microlocal defect measures, Comm. PDEs, 16 (1991), 1761-1794.
[43]	P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-380.
[44]	F. Golse, S. Jin and C. D. Levermore, The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369.
[45]	T. G. Ho, L. J. Landau and A. J. Wilkins, On the weak coupling limit for a Fermi gas in a random potential, Rev. Math. Phys., 5 (1992), 209-298.
[46]	H. Hochstadt, "The Functions of Mathematical Physics," Dover Publications, New York, 1986.
[47]	T. Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 227 (1999), 913-943.
[48]	I. A. Ibragimov and Yu. V. Linnik, "Independent and Stationary Sequences of Random Variables," Wolters-Noordhoff Publishing, Groningen, 1971.
[49]	A. Ishimaru, "Wave Propagation and Scattering in Random Media," New York, Academics, 1978.
[50]	J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes," Grundlehren der mathematischen Wissenschaft 288, Springer-Verlag, 2003.
[51]	A. Jakubowski, A non-Skorohod topology on the Skorohod space, Electron. J. Probability, 2 (1997), 1-21.
[52]	H. Kesten and G. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys., 65 (1979), 97-128.
[53]	H. Kesten and G. C. Papanicolaou, A limit theorem for stochastic acceleration, Comm. Math. Phys., 78 (1980), 19-63.
[54]	R. Khasminskii, A limit theorem for solutions of differential equations with a random right hand side, Theory Probab. Appl., 11 (1966), 390-406.
[55]	T. Komorowski, Diffusion approximation for the advection of particles in a a strongly turbulent random environment, Ann. Probab., 24 (1996), 346-376.
[56]	T. Komorowski, Sz. Peszat and L. Ryzhik, Limit of fluctuations of solutions of Wigner equation, Comm. Math. Phys., 292 (2009), 479-510.
[57]	T. Komorowski and L. Ryzhik, Diffusion in a weakly random Hamiltonian flow, Comm. Math. Phys., 263 (2006), 277-323.
[58]	T. Komorowski and L. Ryzhik, The stochastic acceleration problem in two dimensions, Israel Jour.Math., 155 (2006), 157-204.
[59]	T. Komorowski and L. Ryzhik, Asymptotics of the phase of the solutions of the random Schrödinger equation, preprint, (2010).
[60]	P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618.
[61]	D. Liu, S. Vasudevan, J. Krolik, G. Bal and L. Carin, Electromagnetic time-reversal imaging in changing media: Experiment and analysis, IEEE Trans. Antennas and Prop., 55 (2007), 344-354.
[62]	J. Lukkarinen and H. Spohn, Kinetic limit for wave propagation in a random medium, Arch. Ration. Mech. Anal., 183 (2007), 93-162.
[63]	P. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math., 81 (1999), 595-630.
[64]	P. Markowich, P. Pietra, C. Pohl and H. P. Stimming, A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 1281-1310.
[65]	I. Mitoma, On the sample continuity of $\mathcal S'$ processes, J. Math. Soc. Japan, 35 (1983), 629-636.
[66]	B. Øksendal, "Stochastic Differential Equations," Springer-Verlag, Berlin, 2000.
[67]	G. Papanicolaou and W. Kohler, Asymptotic theory of mixing stochastic ordinary differential equations, Comm. Pure Appl. Math., 27 (1974), 641-668.
[68]	G. Papanicolaou, L. Ryzhik and K. Sølna, The parabolic wave approximation and time reversal, Matematica Contemporanea, 23 (2002), 139-159.
[69]	G. C. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Ito-Schroedinger equation, Multiscale Model. Simul., 6 (2007), 468-492.
[70]	F. Poupaud and A. Vasseur, Classical and quantum transport in random media, J. Math. Pures Appl., 6 (2003), 711-748.
[71]	L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370.
[72]	H. Sato and M. C. Fehler, "Seismic Wave Propagation and Scattering in the Heterogeneous Earth," AIP series in modern acoustics and signal processing, AIP Press, Springer, New York, 1998.
[73]	P. Sheng, "Introduction to Wave Scattering, Localization and Mesoscopic Phenomena," Academic Press, New York, 1995.
[74]	H. Spohn, Derivation of the transport equation for electrons moving through random impurities, Jour. Stat. Phys., 17 (1977), 385-412.
[75]	G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 507-517.
[76]	D. W. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes," Grundlehren der mathematischen Wissenschaften 233, Berlin, Heidelberg, New York, 1979, 1997.
[77]	C. R. Vogel, "Computational Methods for Inverse Problems," Frontiers Appl. Math., SIAM, Philadelphia, 2002.
[78]	E. Wigner, On the quantum correction for thermodynamic equilibrium, Physical Rev., 40 (1932), 749-759.
[79]	B. White, The stochastic caustic, SIMA Jour. Appl. Math., 44 (1984), 127-149.

show all references

References:

[1]	F. Bailly, J. F. Clouet and J.-P. Fouque, Parabolic and gaussian white noise approximation for wave propagation in random media, SIAM J. Appl. Math, 56 (1996), 1445-1470.
[2]	G. Bal, On the self-averaging of wave energy in random media, Multiscale Model. Simul., 2 (2004), 398-420.
[3]	G. Bal, Kinetics of scalar wave fields in random media, Wave Motion, 43 (2005), 132-157.
[4]	G. Bal, Inverse problems in random media: A kinetic approach, J. Phys. Conf. Series, 124 (2008), 012001.
[5]	G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001.
[6]	G. Bal, L. Carin, D. Liu and K. Ren, Experimental validation of a transport-based imaging method in highly scattering environments, Inverse Problems, 26 (2007), 2527-2539.
[7]	G. Bal, D. Liu, S. Vasudevan, J. Krolik and L. Carin, Electromagnetic time-reversal imaging in changing media: Experiment and analysis, IEEE Trans. Anten. and Prop., 55 (2007), 344-354.
[8]	G. Bal, T. Komorowski and L. Ryzhik, Self-averaging of Wigner transforms in random media, Comm. Math. Phys., 242 (2003), 81-135.
[9]	G. Bal, T. Komorowski and L. Ryzhik, Asymptotics of the solutions of the random Schródinger equation, to appear in Arch. Rat. Mech., (2010).
[10]	G. Bal, G. Papanicolaou and L. Ryzhik, Radiative transport limit for the random Schrödinger equations, Nonlinearity, 15 (2002), 513-529.
[11]	G. Bal, G. Papanicolaou and L. Ryzhik, Self-averaging in time reversal for the parabolic wave equation, Stochastics and Dynamics, 4 (2002), 507-531.
[12]	G. Bal and O. Pinaud, Time reversal-based imaging in random media, Inverse Problems, 21 (2005), 1593-1620.
[13]	G. Bal and O. Pinaud, Accuracy of transport models for waves in random media, Wave Motion, 43 (2006), 561-578.
[14]	G. Bal and O. Pinaud, Kinetic models for imaging in random media, Multiscale Model. Simul., 6 (2007), 792-819.
[15]	G. Bal and O. Pinaud, Self-averaging of kinetic models for waves in random media, Kinetic Related Models, 1 (2008), 85-100.
[16]	G. Bal and O. Pinaud, Imaging using transport models for wave-wave correlations, to appear in M3AS, (2011).
[17]	G. Bal and K. Ren, Transport-based imaging in random media, SIAM J. Applied Math., 68 (2008), 1738-1762.
[18]	G. Bal and L. Ryzhik, Time reversal and refocusing in random media, SIAM J. Appl. Math., 63 (2003), 1475-1498.
[19]	G. Bal and L. Ryzhik, Time splitting for wave equations in random media, preprint, (2004).
[20]	G. Bal and L. Ryzhik, Stability of time reversed waves in changing media, Discrete Contin. Dyn. Syst., 12 (2005), 793-815.
[21]	G. Bal and R. Verástegui, Time reversal in changing environment, Multiscale Model. Simul., 2 (2004), 639-661.
[22]	W. Bao, S. Jin and P. A. Markowich, On Time-Splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comp. Phys., 175 (2002), 487-524.
[23]	P. Billingsley, "Convergence of Probability Measures," Wiley, 1999.
[24]	P. Blankenship and G. C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbances. i, SIAM J. Appl. Math., 34 (1978), 437-476.
[25]	P. Blomgren, G. C. Papanicolaou and H. Zhao, Super-Resolution in Time-Reversal Acoustics, J. Acoust. Soc. Am., 111 (2002), 230-248.
[26]	B. Borcea, G. C. Papanicolaou and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems, 19 (2003), S139-S164.
[27]	B. Borcea, G. C. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460.
[28]	B. Borcea, G. C. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems, 22 (2006), 1405-1436.
[29]	A. N. Borodin, A limit theorem for solutions of differential equations with random right hand side, Teor. Veroyatn. Ee Primen, 22 (1977), 498-512.
[30]	R. A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc, 108 (1994), viii+125.
[31]	S. Chandrasekhar, "Radiative Transfer," Dover Publications, New York, 1960.
[32]	G. C. Cohen, "Higher-Order Numerical Methods for Transient Wave Equations," Scientific Computation, Springer Verlag, Berlin, 2002.
[33]	D. A. Dawson and G. C. Papanicolaou, A random wave process, Appl. Math. Optim., 12 (1984), 97-114.
[34]	D. Dürr, S. Goldstein and J. Lebowitz, Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model, Comm. Math. Phys., 113 (1987), 209-230.
[35]	D. R. Durran, "Numerical Methods for Wave equations in Geophysical Fluid Dynamics," Springer, New York, 1999.
[36]	G. F. Edelmann, T. Akal, W. S. Hodgkiss, S. Kim, W. A. Kuperman and H. C. Song, An initial demonstration of underwater acoustic communication using time reversal, IEEE J. Oceanic Eng., 27 (2002), 602-609.
[37]	L. Erdös and H. T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Comm. Pure Appl. Math., 53 (2000), 667-735.
[38]	L. Evans and M. Zworski, Lectures on semiclassical analysis,, Berkeley., ().
[39]	A. Fannjiang, Self-averaging in scaling limits for random high-frequency parabolic waves, Archives of Rational Mechanics and Analysis, 175 (2005), 343-387.
[40]	J. P. Fouque, La convergence en loi pour les processus à valeur dans un espace nucléaire, Ann. Inst. H. Poincaré Prob. Stat, 20 (1984), 225-245.
[41]	J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media," Springer Verlag, New York, 2007.
[42]	P. Gérard, Microlocal defect measures, Comm. PDEs, 16 (1991), 1761-1794.
[43]	P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-380.
[44]	F. Golse, S. Jin and C. D. Levermore, The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369.
[45]	T. G. Ho, L. J. Landau and A. J. Wilkins, On the weak coupling limit for a Fermi gas in a random potential, Rev. Math. Phys., 5 (1992), 209-298.
[46]	H. Hochstadt, "The Functions of Mathematical Physics," Dover Publications, New York, 1986.
[47]	T. Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 227 (1999), 913-943.
[48]	I. A. Ibragimov and Yu. V. Linnik, "Independent and Stationary Sequences of Random Variables," Wolters-Noordhoff Publishing, Groningen, 1971.
[49]	A. Ishimaru, "Wave Propagation and Scattering in Random Media," New York, Academics, 1978.
[50]	J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes," Grundlehren der mathematischen Wissenschaft 288, Springer-Verlag, 2003.
[51]	A. Jakubowski, A non-Skorohod topology on the Skorohod space, Electron. J. Probability, 2 (1997), 1-21.
[52]	H. Kesten and G. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys., 65 (1979), 97-128.
[53]	H. Kesten and G. C. Papanicolaou, A limit theorem for stochastic acceleration, Comm. Math. Phys., 78 (1980), 19-63.
[54]	R. Khasminskii, A limit theorem for solutions of differential equations with a random right hand side, Theory Probab. Appl., 11 (1966), 390-406.
[55]	T. Komorowski, Diffusion approximation for the advection of particles in a a strongly turbulent random environment, Ann. Probab., 24 (1996), 346-376.
[56]	T. Komorowski, Sz. Peszat and L. Ryzhik, Limit of fluctuations of solutions of Wigner equation, Comm. Math. Phys., 292 (2009), 479-510.
[57]	T. Komorowski and L. Ryzhik, Diffusion in a weakly random Hamiltonian flow, Comm. Math. Phys., 263 (2006), 277-323.
[58]	T. Komorowski and L. Ryzhik, The stochastic acceleration problem in two dimensions, Israel Jour.Math., 155 (2006), 157-204.
[59]	T. Komorowski and L. Ryzhik, Asymptotics of the phase of the solutions of the random Schrödinger equation, preprint, (2010).
[60]	P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618.
[61]	D. Liu, S. Vasudevan, J. Krolik, G. Bal and L. Carin, Electromagnetic time-reversal imaging in changing media: Experiment and analysis, IEEE Trans. Antennas and Prop., 55 (2007), 344-354.
[62]	J. Lukkarinen and H. Spohn, Kinetic limit for wave propagation in a random medium, Arch. Ration. Mech. Anal., 183 (2007), 93-162.
[63]	P. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math., 81 (1999), 595-630.
[64]	P. Markowich, P. Pietra, C. Pohl and H. P. Stimming, A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 1281-1310.
[65]	I. Mitoma, On the sample continuity of $\mathcal S'$ processes, J. Math. Soc. Japan, 35 (1983), 629-636.
[66]	B. Øksendal, "Stochastic Differential Equations," Springer-Verlag, Berlin, 2000.
[67]	G. Papanicolaou and W. Kohler, Asymptotic theory of mixing stochastic ordinary differential equations, Comm. Pure Appl. Math., 27 (1974), 641-668.
[68]	G. Papanicolaou, L. Ryzhik and K. Sølna, The parabolic wave approximation and time reversal, Matematica Contemporanea, 23 (2002), 139-159.
[69]	G. C. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Ito-Schroedinger equation, Multiscale Model. Simul., 6 (2007), 468-492.
[70]	F. Poupaud and A. Vasseur, Classical and quantum transport in random media, J. Math. Pures Appl., 6 (2003), 711-748.
[71]	L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370.
[72]	H. Sato and M. C. Fehler, "Seismic Wave Propagation and Scattering in the Heterogeneous Earth," AIP series in modern acoustics and signal processing, AIP Press, Springer, New York, 1998.
[73]	P. Sheng, "Introduction to Wave Scattering, Localization and Mesoscopic Phenomena," Academic Press, New York, 1995.
[74]	H. Spohn, Derivation of the transport equation for electrons moving through random impurities, Jour. Stat. Phys., 17 (1977), 385-412.
[75]	G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 507-517.
[76]	D. W. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes," Grundlehren der mathematischen Wissenschaften 233, Berlin, Heidelberg, New York, 1979, 1997.
[77]	C. R. Vogel, "Computational Methods for Inverse Problems," Frontiers Appl. Math., SIAM, Philadelphia, 2002.
[78]	E. Wigner, On the quantum correction for thermodynamic equilibrium, Physical Rev., 40 (1932), 749-759.
[79]	B. White, The stochastic caustic, SIMA Jour. Appl. Math., 44 (1984), 127-149.

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American Institute of Mathematical Sciences

Kinetic limits for waves in a random medium

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Kinetic limits for waves in a random medium

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