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Limit of Fluctuations of Solutions of Wigner Equation

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Abstract

We consider fluctuations of the solution W ε (t, x, k) of the Wigner equation which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of W ε (t, x, k) converges as ε → 0 to \({\bar{W}(t,x,k)}\) which satisfies the radiative transport equation. We prove that when the initial data is singular in the x variable, that is, W ε (0, x, k) = δ(x)f(k) and \({f\in {\mathcal{S}}(\mathbb{R}^d)}\), then the laws of the rescaled fluctuation \({Z_\varepsilon(t):=\varepsilon^{-1/2}[W_\varepsilon(t,x,k)-\bar{W}(t,x,k)]}\) converge, as ε → 0+, to the solution of the same radiative transport equation but with a random initial data. This complements the result of [6], where the limit of the covariance function has been considered.

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Authors and Affiliations

  1. Institute of Mathematics, UMCS, pl. Marii Curie-Skłodowskiej 1, Lublin, 20-031, Poland

    Tomasz Komorowski

  2. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950, Warsaw, Poland

    Tomasz Komorowski

  3. Institute of Mathematics, Polish Academy of Sciences, Sw. Tomasza 30/7, 31-027, Cracow, Poland

    Szymon Peszat

  4. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Lenya Ryzhik

Authors
  1. Tomasz Komorowski
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  2. Szymon Peszat
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  3. Lenya Ryzhik
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Correspondence to Tomasz Komorowski.

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Communicated by H. Spohn

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Komorowski, T., Peszat, S. & Ryzhik, L. Limit of Fluctuations of Solutions of Wigner Equation. Commun. Math. Phys. 292, 479–510 (2009). https://doi.org/10.1007/s00220-009-0895-0

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  • DOI: https://doi.org/10.1007/s00220-009-0895-0

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