Scaling laws for the multidimensional burgers equation with quadratic external potential

Journal of Statistical Physics

Volumn 124, Issue 1, 2006, Pages 191-205

  Leonenko, N N ^a   Ruiz Medina, M D ^b  

^a CARDIFF UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF GRANADA   (Spain)

Author keywords

Nonhomogeneous multidimensional Burgers equation; Quadratic external potential; Scaling laws; Spatiotemporal random fields; Strongly dependent random initial conditions

Indexed keywords

EID: 33748076806     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-006-9136-5     Document Type: Review

References (33)

1. B. Simon. Functional Integration and Quantum Physics. Academic Press, New York (1979).
2. M. Demuth and J.A. van Casteren. Stochastic. Spectral Theory for Selfadjoint Operators. Birkhauser-Verlag, Basel (2000).
3. M. Rosenblatt. Remarks on Burgers equation. J. Math. Phys. 24:1129-1136 (1968).
4. M. Rosenblatt. Scale renormalization and random solutions of Burgers equation. J. Appl. Prob. 24:328-338 (1987).
5. Ya. G. Sinai. Two results concerning asymptotic behavior of the Burgers equation with force, J. Stat. Phys. 64:1-12 (1992).
6. S. Albeverio, S. A. Molchanov and D. Surgailis. Stratified structure of the Universe and Burgers' equation: A probabilistic approach. Prob. Theor. Rel. Fields 100:457-484 (1994).
7. A. V. Bulinski and S. A. Molchanov. Asymptotic Gaussianess of solutions of the Burgers equation with random initial conditions. Theory Probab. Appl. 36:217-235 (1991).
8. I. Deriev and N. Leonenko. Limit Gaussian behavior of the solutions of the multidimensional Burgers' equation with weak-dependent initial conditions. Acta Applicandae Math. 47:1-18 (1997).
9. N. Leonenko and E. Orsingher. Limit theorems for solutions of Burgers equation with Gaussian and non-Gaussian initial data. Theory Prob. Appl. 40:387-403 (1995).
10. N. N. Leonenko, E. Orsingher and K. V. Rybasov. Limit distributions of solutions of multidimensional Burgers equation with random initial data, I and II. Ukrain. Math. J. 46:870-877 (1994) and 1003-1010.
11. Random Ope. Stoch Equ. 4:229-238.
12. N. N. Leonenko and W. A. Woyczynski. Exact parabolic asymptotics for singular n -D Burgers random fields: Gaussian approximation. Stock. Proc. Appl. 76:141-165 (1998a).
13. N. N. Leonenko and W. A. Woyczynski. Scaling limits of solutions of the heat equation for singular non-Gaussian data, J. Stat. Phys., 91:423-438 (1998b).
14. M. D. Ruiz-Medina, J. M. Angulo and V. V. Anh. Scaling limit solution of a fractional Burgers equation, Stoch. Proc. Appl 93:285-300 (2001).
15. D. Surgailis and W. A. Woyczynski. Limit theorem for the Burgers equation initiated by data with long-range dependence, in Doukhan, P., Oppenheim, G. and Taqqu, M. S. (eds), Theory and Applications of Long-Range Dependence, pp. 507-523. Birkhäuser, Boston (2003).
16. V. V. Anh and N. N. Leonenko. Non-Gaussian scenarios for the heat equation with singular initial conditions. Stock. Proc. Appl. 84:91-114 (1999).
17. V. V. Anh and N. N. Leonenko. Spectral analysis of fractional kinetic equations with random data. J. Statist. Phys. 104:1349-1387 (2001).
18. V. V. Anh and N. N. Leonenko. Renormalization and homogenization of fractional diffusion equations with random data. Prob. Theor. Rel. Fields 124:381-408 (2002).
19. V. V. Anh, N. N. Leonenko and L. M. Sakhno. Spectral properties of Burgers and KPZ turbulence. J. Stat. Physics, in press (2006).
20. V. H. Hoang and K. Khanin. Random Burgers equation and Lagrangian systems in noncompact domains. Nonlinearity 16:819-842 (2003).
21. H. Holden, T. Lindstrom, B. Oksendal, J. Uboe and T. S. Zhang. The Burgers equation with noisy force and the stochastic heat equation. Comm. Partial Diff. Eq. 19:119-141 (1994).
22. d. Ann. Appl. Prob. 7:220-223 (1997).
23. A. I. Saichev and W. A. Woyczynski. Evolution of Burgers turbulence in presence of external forces. J. Fluid Mech. 331:313-343 (1997).
24. W. A. Woyczynski. Burgers-KPZ Turbulence. Göttingen Lectures. Lecture Notes in Mathematics 1706. Springer-Verlag, Berlin (1998).
25. N. Leonenko. Limit Theorems for Random Fields with Singular Spectrum. Kluwer, Dordrecht (1999).
26. O. E. Barndorff-Nielsen and N. N. Leonenko. Burgers turbulence problem with linear or quadratic external potential, J. Appl. Prob. 42:550-565 (2005).
27. L. Bertini and N. Cancrini. The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Physics. 78:1377-1401 (1995).
28. V. V. Anh, J. M. Angulo and M. D. Ruiz-Medina. Possible Long-Range Dependence in Fractional Random Fields. J. Stat. Plan. Inf. 80:95-110 (1999).
29. X. Wang, C. H. Oh and L. C. Kwek. General approach to functional forms for the exponential quadratic operators in coordinate-momentum space. J. Phys. A.: Math. Gen. 31:4329-4336 (1998).
30. E. M. Stein. Singular Integrals and Differential Properties of Functions. Princeton Univerisity Press (1970).
31. A. G. Ramm. Random Fields Estimation Theory, Logman Scientific & Technical (1990).
32. H. Triebel. Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Co (1978).
33. M. D. Ruiz-Medina, J. M. Angulo and V. V. Anh. Fractional generalized random fields on bounded domains. Stock Anal. Appl. 21:465-492 (2003).