Fractional stochastic processes

Fractional stochastic processes and fields and anomalous diffusion models

We introduce and study the stochastic processes and random fields arising in chaotic systems and linear and non-linear and fractal pseudodifferential systems, their spectral theory and related problems. These models have an applications in anomalous diffusions and based on the fractional calculus.

Multifractal systems

We introduce and study the limit theorems for multifractal products of stationary stochastic processes with weak and strong dependence. Some exact form of Renyi function is presented in form of log-scenarios of parabolic and hyperbolic distributions based on the theory of Levy-driven Ornstein-Uhlenback processes and their superpositions.

Cardiff Investigators

• Prof. N. Leonenko

Collaborators

Prof. Murad Taqqu, Prof. Chris Heyde, Prof. Mark Meerschaert, Prof. Vo Anh, Prof. Jose Angulo, Prof. Maria-Dolores Ruiz-Medina, Prof. Narn-Rueih Shieh, Dr. Emanuele Taufer, Dr. Mark Kelbert, Dr. Alla Sikorskii, Dr. Ludmila Sakhno, Dr. Ross McVinish, Dr. Denis Denisov

Selected publications

• Leonenko, N.N., Ruiz-Medina, M.D., Taqqu, M.S. (2011) Fractional elliptic, hyperbolic and parabolic random fields, Electronic Journal of Probability, vol 16, Paper n 40, pages 1134-1172
• Leonenko, N.N., Meerschaert, M.M., and Sikorskii, A. (2011) Fractional Pearson diffusion, Journal of Functional Analysis, submitted
• Liu, F., Meerschaert, M., Momani, Sh., Leonenko, N., Chen W. and Agrawal Om, Eds (2010) "Fractional Differential Equations," Special Issue of the International Journal of Differential Equations.
• Angulo J.M., Kelbert M.Ya., Leonenko, N.N, Ruiz-Medina, M.D. (2008) Spatio-temporal random fields associated with stochastic fractional Helmgoltz and heat equations, Stochastic Environmental Research and Risk Assessment, 22, N1, 3-13
• 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
• 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
• Kelbert, M.Ya., Leonenko, N.N., Ruiz-Medina, M. D., (2005), Fractional random fields associated with stochastic fractional heat equations. Adv. in Appl. Probab. 37, N 1, 108-133
• Anh, V.V., Heyde, C.C., Leonenko, N.N., (2002) Dynamic models of long-memory processes driven by Levy noise. J. Appl. Probab. 39, N 4, 730-747
• Anh, V.V., Leonenko, N.N. (2002) Renormalization and homogenization of fractional diffusion equations with random data. Probab. Theory Related Fields 124, N 3, 381-408
• Anh, V.V., Leonenko, N.N., McVinish, R. (2001) Models for fractional Riesz-Bessel motion and related processes. Fractals. 7, 9, no. 3, 329-346
• Anh, V.V., Leonenko, N.N. (2001) Spectral analysis of fractional kinetic equations with random data. J. Statist. Phys. 104, no. 5-6, 1349-1387
• Anh, V.V., Leonenko, N.N., (1999), Non-Gaussian scenarios for the heat equation with singular initial conditions, Stoch. Proc. Appl, 84, 91-114

• Leonenko, N.N and Shieh N.-R. (2011) Renyi function for multifractal random fields, Fractals, submitted
• Denisov, D., Leonenko, N. (2011) Limit theorems for multifractal products of stationary processes, in preparation. Electronic Journal of Probability, submitted, published in http://arxiv.org/abs/1110.2428
• Anh, V.V., Leonenko, N.N., Shieh, N.-R., Taufer, E. (2010) Simulation of multifractal products of Ornstein-Uhlenbeck type processes. Nonlinearity 23 823--843.
• Anh, V.V., Leonenko, N.N., Shieh, N.-R. (2009) Multifractal scaling of products of birth-death processes. Bernoulli 15 508-531
• Anh, V.V., Leonenko, N.N., Shieh, N.-R. (2009) Multifractal products of stationary diffusion processes. Stoch. Anal. Appl. 27 475-499
• Anh, V.V., Leonenko, N.N., Shieh, N.-R. (2008) Multifractality of products of geometric Ornstein-Uhlenbeck-type processes. Adv. in Appl. Probab. 40 1129--1156