# American Institute of Mathematical Sciences

May  2012, 17(3): 871-914. doi: 10.3934/dcdsb.2012.17.871

## Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential

 1 Institute of Mathematics, UMCS 20-031, Lublin, Poland and IMPAN 00-956 Warsaw, Poland 2 Department of Mathematics, Stanford University, Stanford, CA 94305

Received  July 2010 Revised  February 2011 Published  January 2012

We consider energy fluctuations for solutions of the Schrödinger equation with an Ornstein-Uhlenbeck random potential when the initial data is spatially localized. The limit of the fluctuations of the Wigner transform satisfies a kinetic equation with random initial data. This result generalizes that of [12] where the random potential was assumed to be white noise in time.
Citation: Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871
##### References:
 [1] G. Bal, G. Papanicolaou and L. Ryzhik, Radiative transport limit for the random Schroedinger equation,, Nonlinearity, 15 (2002), 513. doi: 10.1088/0951-7715/15/2/315. Google Scholar [2] 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. doi: 10.1088/0266-5611/23/6/015. Google Scholar [3] 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. doi: 10.1109/TAP.2006.889807. Google Scholar [4] G. Bal and O. Pinaud, Kinetic models for imaging in random media,, Multiscale Model. Simul., 6 (2007), 792. doi: 10.1137/060678464. Google Scholar [5] G. Bal and O. Pinaud, Self-averaging of kinetic models for waves in random media,, Kinet. Relat. Models, 1 (2008), 85. doi: 10.3934/krm.2008.1.85. Google Scholar [6] G. Bal and O. Pinaud, Dynamics of wave scintillation in random media,, Comm. PDEs, 35 (2010), 1176. doi: 10.1080/03605301003801557. Google Scholar [7] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications, 44 (1992). Google Scholar [8] 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. Google Scholar [9] A. Fannjiang, Self-averaging scaling limits for random parabolic waves,, Arch. Rational Mech. Anal., 175 (2005), 343. doi: 10.1007/s00205-004-0343-4. Google Scholar [10] P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Comm. Pure Appl. Math., 50 (1997), 323. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C. Google Scholar [11] C. Kipnis and C. Landim, "Scaling Limits of Interacting Particle Systems,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 320 (1999). Google Scholar [12] T. Komorowski, Sz. Peszat and L. Ryzhik, Limit of fluctuations of solutions of Wigner equation,, Comm. Math. Phys., 292 (2009), 479. doi: 10.1007/s00220-009-0895-0. Google Scholar [13] M. Ledoux and M. Talagrand, "Probability in Banach Spaces. Isoperimetry and Processes,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23 (1991). Google Scholar [14] I. Mitoma, Tightness of probabilities On $C([ 0, 1 ]; \mathcalS)$ and $D([ 0, 1 ]; \mathcalS)$,, Ann. Probab., 11 (1983), 989. doi: 10.1214/aop/1176993447. Google Scholar [15] L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media,, Wave Motion, 24 (1996), 327. doi: 10.1016/S0165-2125(96)00021-2. Google Scholar [16] H. Spohn, Derivation of the transport equation for electrons moving through random impurities,, Jour. Stat. Phys., 17 (1977), 385. doi: 10.1007/BF01014347. Google Scholar

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##### References:
 [1] G. Bal, G. Papanicolaou and L. Ryzhik, Radiative transport limit for the random Schroedinger equation,, Nonlinearity, 15 (2002), 513. doi: 10.1088/0951-7715/15/2/315. Google Scholar [2] 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. doi: 10.1088/0266-5611/23/6/015. Google Scholar [3] 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. doi: 10.1109/TAP.2006.889807. Google Scholar [4] G. Bal and O. Pinaud, Kinetic models for imaging in random media,, Multiscale Model. Simul., 6 (2007), 792. doi: 10.1137/060678464. Google Scholar [5] G. Bal and O. Pinaud, Self-averaging of kinetic models for waves in random media,, Kinet. Relat. Models, 1 (2008), 85. doi: 10.3934/krm.2008.1.85. Google Scholar [6] G. Bal and O. Pinaud, Dynamics of wave scintillation in random media,, Comm. PDEs, 35 (2010), 1176. doi: 10.1080/03605301003801557. Google Scholar [7] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications, 44 (1992). Google Scholar [8] 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. Google Scholar [9] A. Fannjiang, Self-averaging scaling limits for random parabolic waves,, Arch. Rational Mech. Anal., 175 (2005), 343. doi: 10.1007/s00205-004-0343-4. Google Scholar [10] P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Comm. Pure Appl. Math., 50 (1997), 323. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C. Google Scholar [11] C. Kipnis and C. Landim, "Scaling Limits of Interacting Particle Systems,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 320 (1999). Google Scholar [12] T. Komorowski, Sz. Peszat and L. Ryzhik, Limit of fluctuations of solutions of Wigner equation,, Comm. Math. Phys., 292 (2009), 479. doi: 10.1007/s00220-009-0895-0. Google Scholar [13] M. Ledoux and M. Talagrand, "Probability in Banach Spaces. Isoperimetry and Processes,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23 (1991). Google Scholar [14] I. Mitoma, Tightness of probabilities On $C([ 0, 1 ]; \mathcalS)$ and $D([ 0, 1 ]; \mathcalS)$,, Ann. Probab., 11 (1983), 989. doi: 10.1214/aop/1176993447. Google Scholar [15] L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media,, Wave Motion, 24 (1996), 327. doi: 10.1016/S0165-2125(96)00021-2. Google Scholar [16] H. Spohn, Derivation of the transport equation for electrons moving through random impurities,, Jour. Stat. Phys., 17 (1977), 385. doi: 10.1007/BF01014347. Google Scholar
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