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

show all references

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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