Skip to main content
SpringerLink
Log in

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

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • S. Albeverio, S. A. Molchanov, and D. Surgailis, “Stratified structure of the Universe and Burgers’ equation: A probabilistic approach,” Probability Theory and Related Fields vol. 100 pp. 457–484, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  • J. M. Angulo, V. V. Anh, R. McVinish, and M. D. Ruiz-Medina, “Fractional kinetic equation driven by Gaussian or infinitely divisible noise,” Advances in Applied Probability vol. 37 pp. 366–392, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  • J. M. Angulo, M. D. Ruiz-Medina, V. V. Anh, and W. Grecksch, “Fractional diffusion and fractional heat equation,” Advances in Applied Probability vol. 32 pp. 1077–1099, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  • V. V. Anh, and N. N. Leonenko, “Non-Gaussian scenarios for the heat equation with singular initial conditions,” Stochastic Processes and their Applications vol. 84 pp. 91–114, 1999.

    Article  MathSciNet  Google Scholar 

  • V. V. Anh, and N. N. Leonenko, “Renormalization and homogenization of fractional diffusion equations with random data,” Probability Theory and Related Fields vol. 124 pp. 381–408, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  • V. V. Anh, N. N. Leonenko, and L. M. Sakhno, “Spectral properties of Burgers and KPZ turbulence,” Journal of Statistical Physics 112, 949–974, 2006.

    Article  MathSciNet  Google Scholar 

  • O. E. Barndorff-Nielsen, and N. N. Leonenko, “Burgers turbulence problem with linear or quadratic external potential,” Journal of Applied Probabability vol. 42(N2) pp. 550–565, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  • N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag: Berlin, 1992.

    MATH  Google Scholar 

  • L. Bertini, and N. Cancrini, “The stochastic heat equation: Feynman-Kac formula and intermittence,” Journal of Statistical Physics vol. 78(N 5/6) pp. 1377–1401, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • P. Breuer, and P. Major, “Central limit theorem for non-linear functionals of Gaussian fields,” Journal of Multivariate Analysis vol. 13 pp. 425–441, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  • A. V. Bulinski, and S. A. Molchanov, “Asymptotic Gausianness of solutions of the Burgers equation with random initial conditions,” Theor. Prob. Appl. vol. 36 pp. 217–235, 1991.

    Article  Google Scholar 

  • J. Burgers, The Nonlinear Diffusion Equation, Kluwer: Dordrecht, 1974.

    MATH  Google Scholar 

  • D. Chambers, and E. Slud, “Central limit theorem for nonlinear functionals of stataionary Gaussian processes,” Probability Theory and Related Fields vol. 80, pp. 323–346, 1989.

    Article  MathSciNet  Google Scholar 

  • A. J. Chorin, Lecture Notes in Turbulence Theor. Publish or Perish, Berkeley, CA, 1975.

  • H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators, Springer-Verlag: Berlin, 1987.

    MATH  Google Scholar 

  • M. Demuth, and J. A. van Casteren, Stochastic Spectral Theory for Selfadjoint Operators, Birkhauser-Verlag: Basel, 2000.

    MATH  Google Scholar 

  • I, Deriev, and N. Leonenko, “Limit Gaussian behavior of the solutions of the multidimensional Burgers’ equation with weak-dependent initial conditions,” Acta Applicandae Mathematicae vol. 47 pp. 1–18, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Dermone, S. Hamadene, and Y. Ouknine, “Limit theorems for statistical solution of Burgers equation,” Stochastic Processes and their Applications vol. 81, pp. 17–230, 1999.

    Google Scholar 

  • R. L. Dobrushin, and P. Major, “Non-central limit theorems for nonlinear fuctionals of Gaussian fields,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete vol. 50 pp. 1-28, 1979.

    Article  MathSciNet  Google Scholar 

  • P. Doukhan, G. Oppenheim, and M. S. Taqqu, Theory and Applications of Long-Range Dependence, Birkhauser: Boston, 2003.

    MATH  Google Scholar 

  • T. Funaki, D. Surgailis, and W. A. Woyczynski, “Gibbs–Cox random fields and Burgers turbulence,” Annals Applied Probability vol. 5 pp. 461–492, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • S. Gurbatov, A. Malakhov, and A. Saichev, Non-linear Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles, Manchester University Press: Manchester, 1991.

    Google Scholar 

  • M. Holden, B. Øksendal, J. Ubøe, and T. S. Zhang, Stochastic Partial Differential Equations: A Modelling, White Noise Functional Approach, Birkhäuser: Boston, 1996.

    Google Scholar 

  • E. Hopf, “The partial differential equation u x  + uu x  = μu xx ,” Communications on Pure Applied Mathematics vol. 3 pp. 201–230, 1950.

    Article  MATH  MathSciNet  Google Scholar 

  • I, Iribarren, and J. R. León, “Central limit theorem for solutions of random initialized differential equations: A simple proof,” Journal of Applied Mathematics and Stochastic Analysis, doi:10.1155/JAMSA/2006/35206.

  • K. Ishiyama, “Methods for evaluating density functions of exponential functionals represented as integrals of geometric Brownian motion,” Methodology and Computing in Applied Probability vol. 7 pp. 271–283, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  • A. V. Ivanov, and N. N.Leonenko, Statistical Analysis of Random Fields, Kluwer: Dordrecht, 1989.

    MATH  Google Scholar 

  • M. Kelbert, N. Leonenko, and M. D. Ruiz-Medina, “Fractional random fields associated with stochastic fractional heat equations,” Advances in Applied Probability vol. 37 pp. 108–133, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  • S. Kochmanski, “On the evolution operators for some equations of mathematical physics with variable coefficients,” Ukrainian Mathematical Journal vol. 46 pp. 938–952, 1994.

    Article  MathSciNet  Google Scholar 

  • N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum, Kluwer: Dordrecht, 1999.

    MATH  Google Scholar 

  • N. N. Leonenko, and O. A. Melnikova, “Renormalization and homogenization of solutions of heat equation with linear potential and related Burgers equation with random data,” Theor. Prob. Math. Stat. vol. 62 pp. 72–82, 2000.

    MATH  MathSciNet  Google Scholar 

  • N. Leonenko, and E. Orsingher, “Limit theorems for solutions of Burgers equation with Gaussian and non-Gaussian initial data,” Theor. Prob. Appl. vol. 40 pp. 387–403, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • N. N. Leonenko, and M. D. Ruiz-Medina, “Scaling laws for the multidimansional Burgers equation with quadratic external potential,” Journal of Statistical Physics vol. 124 pp. 191–205, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  • N. N. Leonenko, and M. D. Ruiz-Medina, “Gaussian scenario for the heat and Burgers equations with quadratic potential and weakly dependent data,” Journal of Statistical Physics (in process).

  • N. N. Leonenko, and W. A. Woyczynski, “Exact parabolic asymptotics for singular \(n-\mathcal{D}\) Burgers random fields: Gaussian approximation,” Stochastic Processes and their Applications vol. 76 pp. 141–165, 1998a.

    Article  MATH  MathSciNet  Google Scholar 

  • N. N. Leonenko, and W. A. Woyczynski, “Scaling limits of solutions of the heat equation for singular non-Gaussian data,” Journal of Statistical Physics vol. 91, pp. 423–438, 1998b.

    Article  MATH  MathSciNet  Google Scholar 

  • S. A. Molchanov, D. Surgailis, and W. A. Woyczynski, “Hyperbolic asymptotics in Burgers turbulence,” Communications in Mathematical Physics vol. 168 pp. 209–226, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • S. A. Molchanov, D. Surgailis, and W. A. Woyczynski, “The large-scale structure of the Universe and quasi-Voronoi tessellation of shock fronts in forced Burgers turbulence in R d,” Annals of Applied Probability vol. 7 pp. 220–223, 1997.

    MathSciNet  Google Scholar 

  • D. Nualart, and G. Peccati, “Central limit theorems for sequences of multiple stochastic integrals. Convergence in law of multiple stochastic integrals,” Annals of Probability vol. 33 pp. 177-193, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Peccati, and C. A. Tudor, “Gaussian limits for vector-valued multiple stochastic integrals,” Lectures Notes in Math. Séminaire de Probabilités XXXVIII, pp. 247–262, 2005.

  • M. Rosenblatt, “Scale renormalization and random solutions of Burgers equation,” Journal of Applied Probability vol. 24 pp. 328–338, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  • M. D. Ruiz-Medina, J. M. Angulo, and V. V. Anh, “Scaling limit solution of a fractional Burgers equation,” Stochastic Processes and their Applications vol. 93 pp. 285–300, 2001.

    Article  MathSciNet  Google Scholar 

  • B. Simon, Functional Integration and Quantum Physics, Academic Press: New York, 1979.

    MATH  Google Scholar 

  • M. S. Taqqu, “Weak convergence to fractional Brownian motion and to the Rosenblatt process,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete vol. 31 pp. 287–302, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  • M. S. Taqqu, “Convergence of integrated processes of arbitrary Hermite rank,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete vol. 40 pp. 203–238, 1979.

    Article  MathSciNet  Google Scholar 

  • G. B. Witham, Linear and Nonlinear Waves, Wiley: New York, 1974.

    Google Scholar 

  • W. A. Woyczynski, Burgers-KPZ Turbulence, Göttingen Lectures. Lecture Notes in Mathematics vol. 1706. Springer-Verlag: Berlin, 1998.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4YH, UK

    N. N. Leonenko

  2. Department of Statistics and Operations Research, University of Granada, Campus Fuente Nueva s/n, 18071, Granada, Spain

    M. D. Ruiz-Medina

Authors
  1. N. N. Leonenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. D. Ruiz-Medina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. D. Ruiz-Medina.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Leonenko, N.N., Ruiz-Medina, M.D. Gaussian Scenario for the Heat Equation with Quadratic Potential and Weakly Dependent Data with Applications. Methodol Comput Appl Probab 10, 595–620 (2008). https://doi.org/10.1007/s11009-007-9069-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-007-9069-8

Keywords

AMS 2000 Subject Classification

Search

Navigation