Limit theorems

We study classical and non-classical limit theorems for stochastic processes and random fields with weak and strong dependence, as well as Tauberian and Abelian theorems for the correlation functions and spectral densities of a homogeneous isotropic random fields.

Collaborators: Prof. Florin Avram Prof. Vo Anh Prof. Wojbor Woyczynski Prof. Maria Dolores Ruiz-Medina Dr. Emanuele Taufer Dr. Ludmila Sakhno Dr. Andryi Olenko

Burgers turbulence problem

The Burgers equation constitutes an important example of nonlinear partial differential equations studied in turbulence. We study the macro scaling scenarios for the Burgers equation with random initial conditions and with or without quadratic external potential. These scenarios are in some sense subordinated to the Gaussian white noise random measure.

Collaborators: Prof. Ole Barndorff-Nielsen, Prof. Wojbor Woyczynski, Prof. Vo Anh, Prof. Maria-Dolores Ruiz-Medina, Dr. Ludmila Sakhno

Cardiff Investigators

• Prof. Nikolai Leonenko

Selected publications

• Ivanov A.V., Leonenko N.N, Ruiz-Medina, M.D., Savich, I.N. (2011) Limit theorems for weighted non-linear transformations of Gaussian processes with singular spectra, Annals of Probability, submitted revised version.
• Leonenko N.N., Olenko, A. (2011) Toward a computational approach to Tauberian and Abelian theorems, Methodology and Computing in Applied Probability, submitted
• Anh, V.V., Leonenko, N.N., Ruiz-Medina, M.D. (2011) Macroscaling limit theorems for filtered spatiotemporal random fields, Probab. Theory Related Fields , submitted
• Avram, F., Leonenko, N.N, and Sakhno, L. (2010) On Szego type limit theorem, the Holder -Young-Brascamp-Lieb inequality, and asymptotic theory of integrals and quadratic forms of stationary fields, ESAIM: Probablity and Statistics, vol 14, 2010, 210-255
• Anh, V.V., Leonenko, N.N. (2002) Renormalization and homogenization of fractional diffusion equations with random data. Probab. Theory Related Fields 124 (2002), N 3, 381-408
• Anh, V.V., Leonenko, N.N. (1999) Non-Gaussian scenarios for the heat equation with singular initial conditions. Stochastic Process. Appl. 84 (1999), no. 1, 91-114
• Leonenko, N.N., Woyczynski, W.A. (1998) Exact parabolic asymptotics for singular n-D Burgers' random fields: Gaussian approximation. Stochastic Process. Appl. 76 (1998), no. 2, 141-165
• Leonenko, N.N. (1999) Limit Theorems for Random Fields with Singular Spectrum, Mathematics and its Applications, 465. Kluwer Academic Publishers, Dordrecht/Boston/London.
• Leonenko, N.N., Olenko, A.Ya. (1992) Tauberian and Abelian theorems for the correlation function of a homogeneous isotropic random field. Ukrainian Math. J. 43, no. 12, 1539--1548
• Ivanov, A.V., Leonenko, N.N. (1989), Statistical Analysis of Random Fields, Mathematics and its Applications, 28. Kluwer Academic Publishers, Dordrecht/Boston/London

• Leonenko, N.N., Ruiz-Medina, M.D (2011) Random fields arising in chaotic systems: Burgers equation and fractal pseudodifferential systems. In "Space-Time Processes and Challengers Related to Environmental Problems", Lecture Notes in Statistics, vol 207, Springer-Verlag, E.Porcu et al.(Eds), in press
• Leonenko N.N., Ruiz-Medina, M.D.(2010) Spatial scaling for randomly initialized heat and Burgers equation with quadratic potential, Stochastic Analysis and Applications, 28, 303-321
• Leonenko, N.N., Ruiz-Medina, M.D.(2008)Gaussian scenario for the heat and Burgers equations with quadratic external potential and weakly dependent data with applications, Methodology and Computing in Applied Probability, 10, N 4, 595-620
• Anh, V.V., Leonenko, N.N., Sakhno, L.M. (2006) Spectral properties of Burgers and KPZ turbulence. J. Statist. Physics, 122, N 5, 949-974
• Leonenko, N.N, Ruiz-Medina, M.D. (2006) Scaling laws for the multidimensional Burgers equation with quadratic external potential, J. Statist. Physics, 124, N1, 191-205
• Barndorff-Nielsen, O.E., Leonenko, N.N., (2005) Burgers' turbulence problem with linear or quadratic external potential. J. Appl. Probab. 42, N 2, 550-565
• Leonenko, N.N., Woyczynski, W. A. (1999) Parameter identification for singular random fields arising in Burgers' turbulence. J. Statist. Plann. Inference 80, no. 1-2, 1-13
• Leonenko, N.N., Woyczynski, W.A. (1998) Exact parabolic asymptotics for singular n-D Burgers' random fields: Gaussian approximation. Stoch. Proc. Appl., 76, 141-165