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Gaussian Scenario for the Heat Equation with Quadratic Potential and Weakly Dependent Data with Applications

Author

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  • N. N. Leonenko

    (Cardiff University)

  • M. D. Ruiz-Medina

    (University of Granada)

Abstract

For a suitable scaling of the solution to the one-dimensional heat equation with spatial-dependent coefficients and weakly dependent random initial conditions, the convergence to the Gaussian limiting distribution is proved. The scaling proposed and methodology followed allow us to obtain Gaussian scenarios for related equations such as the one-dimensional Burgers equation as well as for the multidimensional formulation of both the heat and Burgers equations. Furthermore, the investigation of non-Gaussian scenarios is opened with a different proposed scaling, proving the convergence of the second-order moments.

Suggested Citation

  • N. N. Leonenko & M. D. Ruiz-Medina, 2008. "Gaussian Scenario for the Heat Equation with Quadratic Potential and Weakly Dependent Data with Applications," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 595-620, December.
    • Handle: RePEc:spr:metcap:v:10:y:2008:i:4:d:10.1007_s11009-007-9069-8
    DOI: 10.1007/s11009-007-9069-8
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    References listed on IDEAS

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    1. Anh, V. V. & Leonenko, N. N., 1999. "Non-Gaussian scenarios for the heat equation with singular initial conditions," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 91-114, November.
    2. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    3. Ruiz-Medina, M. D. & Angulo, J. M. & Anh, V. V., 2001. "Scaling limit solution of a fractional Burgers equation," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 285-300, June.
    4. Kazuyuki Ishiyama, 2005. "Methods for Evaluating Density Functions of Exponential Functionals Represented as Integrals of Geometric Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 7(3), pages 271-283, September.
    5. Dermoune, A. & Hamadène, S. & Ouknine, Y., 1999. "Limit theorem for the statistical solution of Burgers equation," Stochastic Processes and their Applications, Elsevier, vol. 81(2), pages 217-230, June.
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