Abstract
Limiting distributions of the parabolically rescaled solutions of the heat equation with singular non-Gaussian initial data with long-range dependence are described in terms of their multiple stochastic integral representations.
This is a preview of subscription content, log in via an institution to check access.
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, Prob. Theory and Rel. Fields 100:457–484 (1994).
G. A. Bécus, Variational formulation of some problems for the random heat equation, Applied Stochastic Processes, G. Adomian, ed. (Academic Press, New York, 1980), pp. 19–36.
S. Berman, High level sojourns for strongly dependent Gaussian processes, Z. Wahrschein. verw. Gebiete 50:223–236 (1979).
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation (Cambridge University Press, Cambridge, 1987).
P. Breuer and P. Major, Central limit theorem for non-linear functionals of Gaussian fields, J. Multivariate Anal. 13:425–441 (1983).
A. V. Bulinskii and S. A. Molchanov, Asymptotic Gaussianness of solutions of the Burgers equation with random initial data, Theory Prob. Appl. 36:217–235 (1991).
R. A. Carmona and S. A. Molchanov, Stationary parabolic Anderson model and intermittences, Probab. Th. Related Fields 102:433–453 (1995).
R. L. Dobrushin, Gaussian and their subordinated self-similar random generalized fields, Ann. Prob. 7:1–28 (1979).
R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gaussían fields, Z. Wahr. verw. Gebiete 50:1–28 (1979).
T. Funaki, D. Surgailis, and W. A. Woyczynski, Gibbs-Cox random fields and Burgers turbulence, Ann Appl. Prob. 5:461–492 (1995).
H. Holden, B. Øksendal, J. Ubøe, and T.-S. Zhang, Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach (Birkhäuser-Boston, 1996).
Y. Hu and W. A. Woyczynski, An extremal rearrangement property of statistical solutions of the Burgers equation, Ann. Appl. Probab. 4:838–858 (1994).
A. V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields (Kluwer, Dordrecht, 1989).
J. Kampé de Feriet, Random solutions of the partial differential equations, Proc. 3rd Berkeley Symp. Math. Stat. Prob. (University of California Press, Berkeley, California, 1955), Vol. III, pp. 199–208.
S. Kwapien and W. A. Woyczynski, Random Series and Stochastic Integrals: Single and Multiple (Birkhäuser-Boston, 1992).
N. N. Leonenko and A. Ya. Olenko, Tauberian and Abelian theorems for correlation function of homogeneous isotropic random fields, Ukrainian Math. J. 43:1652–1664 (1991).
N. N. Leonenko and I. I. Deriev, Limit theorem for solutions of multidimensional Burgers' equation with weak dependent random initial conditions, Theory Prob. and Mathem. Stat. 51:103–115 (1994).
N. N. Leonenko and E. Orsingher, Limit theorems for solutions of Burgers equation with Gaussian and non-Gaussian initial data, Theory Prob. Appl. 40:387–336 (1995).
P. Major, Multiple Wiener-Ito integrals, Springer's Lecture Notes in Mathematics, Vol. 849 (1981).
S. A. Molchanov, D. Surgailis, and W. A. Woyczynski, Hyperbolic asymptotics in Burgers turbulence, Comm. Math. Phys. 168:209–226 (1995).
J. M. Noble, Evolution equation with Gaussian potential, Nonlinear Analysis, Theory, Methods and Appl. 28:103–135 (1997).
D. Nualart and M. Zakai, Generalized Brownian functionals and the solution to a stochastic partial differential equation, J. Func. Anal. 84:279–296 (1989).
M. Rosenblatt, Remark on the Burgers equation, J. Math. Phys. 9:1129–1136 (1968).
M. Rosenblatt, Independence and dependence, Proc. 4th Berkeley Symp. Math. Stat. Probab. (University of California Press, Berkeley, 1961), pp. 411–443.
N. E. Rotanov, A. G. Shuhov, and Yu. M. Suhov, Stabilization of the statistical solutions of the parabolic equation, Acta Appl. Math. 22:103–115 (1991).
D. Surgailis and W. A. Woyczynski, Scaling limits of solutions of the Burgers equation with singular Gaussian initial data, in Chaos Expansions, Multiple Wiener-Itô Integrals and Their Applications, C. Houdré and V. Pèrez-Abreu, eds. (CRC Press, Boca Raton, FL, 1994).
M. S. Taqqu, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrschein. verw. Gebiete 31:287–302 (1975).
M. S. Taqqu, Convergence of integrated precesses of arbitrary Hermite rank, Z. Wahr. verw. gebiete 50:351–362 (1979).
J. Ubøe and T. Zhang, A stability property of the stochastic heat equation, Stoch. Proc. Appl. 60:247–260 (1995).
D. V. Widder, The Heat Equation (Academic Press, New York, 1975).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Leonenko, N.N., Woyczynski, W.A. Scaling Limits of Solutions of the Heat Equation for Singular Non-Gaussian Data. Journal of Statistical Physics 91, 423–438 (1998). https://doi.org/10.1023/A:1023060625577
Issue Date:
DOI: https://doi.org/10.1023/A:1023060625577