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Burgers' turbulence problem with linear or quadratic external potential

Part of: Inference from stochastic processes Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  14 July 2016

O. E. Barndorff-Nielsen  and
N. N. Leonenko
O. E. Barndorff-Nielsen*
Affiliation:
University of Aarhus
N. N. Leonenko*
Affiliation:
Cardiff University
*
∗Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, K-8000 Aarhus C, Denmark. Email address: oebn@imf.au.dk
∗∗Postal address: Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK. Email address: leonenkon@cardiff.ac.uk
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Abstract

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We consider solutions of Burgers' equation with linear or quadratic external potential and stationary random initial conditions of Ornstein-Uhlenbeck type. We study a class of limit laws that correspond to a scale renormalization of the solutions.

Keywords

Inhomogeneous Burgers equationOrnstein-Uhlenbeck-type processlinear potentialquadratic potentialrandom initial conditionstationary processshort-range dependencelong-range dependenceinfinitely divisible distributionnormal inverse Gaussian distribution

MSC classification

Primary: 62M40: Random fields; image analysis 62M15: Spectral analysis 60H15: Stochastic partial differential equations 60G60: Random fields
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 550 - 565
Copyright
© Applied Probability Trust 2005 

References

Albeverio, S., Molchanov, S. A. and Surgailis, D. (1994). Stratified structure of the universe and Burgers' equation—a probabilistic approach. Prob. Theory Relat. Fields 100, 457–484.Google Scholar
Anh, V. V. and Leonenko, N. N. (1999). Non-Gaussian scenarios for the heat equation with singular initial conditions. Stoch. Process Appl. 84, 91–114.Google Scholar
Barndorff-Nielsen, O. E. (1998a). Probability and statistics, self-decomposability, finance and turbulence. In Probability Towards 2000 (Lecture Notes Statist. 128), eds Accardi, L. and Heyde, C. C., Springer, New York, pp. 47–57.Google Scholar
Barndorff-Nielsen, O. E. (1998b). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein–Uhlenbeck type processes. Theory Prob. Appl. 45, 175–194.Google Scholar
Barndorff-Nielsen, O. E. and Pérez-Abreu, V. (1999). Stationary and self-similar processes driven by Lévy processes. Stoch. Process Appl. 84, 357–369.Google Scholar
Batchelor, G. (1953). The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Berline, N., Getzler, E. and Vergne, M. (1992). Heat Kernels and Dirac Operators. Springer, Berlin.Google Scholar
Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman–Kac formula and intermittence. J. Statist. Phys. 78, 1377–1401.Google Scholar
Bertoin, J. (1998a). Large-deviation estimates in Burgers turbulence with stable noise initial data. J. Statist. Phys. 91, 655–667.Google Scholar
Bertoin, J. (1998b). The inviscid Burgers equation with Brownian initial velocity. Commun. Math. Phys. 193, 397–406.Google Scholar
Bulinskiı˘, A. V. and Molchanov, S. A. (1991). Asymptotic Gaussian property of the solution of the Burgers equation with random initial data. Theory Prob. Appl. 36, 217–236.Google Scholar
Burgers, J. (1974). The Nonlinear Diffusion Equation. Kluwer, Dordrecht.Google Scholar
Chorin, A. J. (1975). Lectures on Turbulence Theory. Publish or Perish, Berkeley, CA.Google Scholar
Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B. (1987). Schrödinger Operators. Springer, Berlin.Google Scholar
Deriev, I. and Leonenko, N. (1997). Limit Gaussian behavior of the solutions of the multidimensional Burgers' equation with weak-dependent initial conditions. Acta Appl. Math. 47, 1–18.Google Scholar
Dermone, A., Hamadene, S. and Ouknine, Y. (1999). Limit theorems for statistical solution of Burgers equation. Stoch. Process. Appl. 81, 17–230.Google Scholar
Dobrushin, R. L. and Major, P. (1979). Major non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrscheinlichkeitsth. 50, 27–52.Google Scholar
Frisch, U. (1995). Turbulence. Cambridge University Press.Google Scholar
Funaki, T., Surgailis, D. and Woyczyński, W. A. (1995). Gibbs–Cox random fields and Burgers turbulence. Ann. Appl. Prob. 5, 461–492.Google Scholar
Gurbatov, S., Malakhov, A. and Saichev, A. (1991) Non-linear Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles. Manchester University Press.Google Scholar
Hodges, S. and Carverhill, A. (1993). Quasi mean reversion in an efficient stock market: the characterization of economic equilibria which support Black–Scholes option pricing. Econom. J. 103, 395–405.Google Scholar
Hodges, S. and Selby, M. J. P. (1997). The risk premium in trading equilibria which support Black–Scholes option pricing. In Mathematics of Derivative Securities, eds Dempster, M. and Pliska, S., Cambridge University Press, pp. 41–53.Google Scholar
Holden, M., Øksendal, B., Ubøe, J. and Zhang, T. S. (1996). Stochastic Partial Differential Equations: A Modelling, White Noise Functional Approach. Birkhäuser, Boston, MA.Google Scholar
Hopf, E. (1950). The partial differential equation ux + uux = μuxx . Commun. Pure Appl. Math. 3, 201–230.Google Scholar
Hu, Y. and Woyczyński, W. A. (1995). Limiting behaviour of quadratic forms of moving averages and statistical solutions of the Burgers' equation. J. Multivariate Anal. 52, 15–44.Google Scholar
Kardar, M., Parisi, G. and Zhang, Y. C. (1986). Dynamical scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892.Google Scholar
Kochmanski, S. (1994). On the evolution operators for some equations of mathematical physics with variable coefficients. Ukranian Math. J. 46, 938–952.CrossRefGoogle Scholar
Leonenko, N. (1999). Limit Theorems for Random Fields with Singular Spectrum. Kluwer, Dordrecht.Google Scholar
Leonenko, N. N. and Li, Z. B. (1994). Non-Gaussian limit distributions of solutions of Burgers equation with strongly dependent random initial conditions. Random Oper. Stoch. Equations 2, 95–102.Google Scholar
Leonenko, N. N. and Meĺnikova, O. O. (2001). Rescaling and homogenization of solutions of the heat equation with a linear potential and of the corresponding Burgers equation with random data. Theory Prob. Math. Statist. 62, 77–88.Google Scholar
Leonenko, N. and Orsingher, E. (1995). Limit theorems for solutions of Burgers equation with Gaussian and non-Gaussian initial data. Theory Prob. Appl. 40, 387–403.Google Scholar
Leonenko, N. N. and Woyczyński, W. A. (1998). Exact parabolic asymptotics for singular n-D Burgers' random fields: Gaussian approximation. Stoch. Process. Appl. 76, 141–165.Google Scholar
Molchanov, S. A., Surgailis, D. and Woyczyński, W. A. (1995). Hyperbolic asymptotics in Burgers turbulence. Commun. Math. Phys. 168, 209–226.Google Scholar
Molchanov, S. A., Surgailis, D. and Woyczyński, W. A. (1997). The large-scale structure of the universe and quasi-Voronoi tessellation of shock fronts in forced Burgers turbulence in Rd . Ann. Appl. Prob. 7, 220–223.Google Scholar
Rosenblatt, M. (1987). Scale renormalization and random solutions of Burgers equation. J. Appl. Prob. 24, 328–338.Google Scholar
Ryan, R. (1998). The statistics of Burgers turbulence initiated with fractional Brownian noise data. Commun. Math. Phys. 191, 1008–1038.Google Scholar
Saichev, A. I. and Woyczyński, W. A. (1997). Evolution of Burgers' turbulence in the presence of external forces. J. Fluid Mech. 331, 313–343.Google Scholar
Shandarin, S. F. and Zeldovich, Ya. B. (1989). Turbulence, intermittency, structures in a left-gravitating medium: the large scale structure of the universe. Rev. Modern Phys. 61, 185–220.Google Scholar
Sinai, Ya. G. (1992). Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148, 601–621.Google Scholar
Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287–302.Google Scholar
Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitsth. 50, 53–83.Google Scholar
Watson, G. N. (1944). A Treatise of the Theory of Bessel Functions. Cambridge University Press.Google Scholar
Witham, G. B. (1974). Linear and Nonlinear Waves. John Wiley, New York.Google Scholar
Woyczyński, W. A. (1998). Burgers-KPZ Turbulence (Lecture Notes Math. 1706). Springer, Berlin.Google Scholar