A generalized Cauchy process and its application to relaxation phenomena. (English) Zbl 1090.82013

Lim, S. C.; Li, Ming

J. Phys. A, Math. Gen. 39, No. 12, 2935-2951 (2006).

The authors introduce a two-parameter family of non-Markovian stationary Gaussian processes whose defining property is that their covariance takes the functional form of the symmetric stable distribution. Processes are not self-similiar. They are unrelated to so-called fractional Brownian motions. Naming these processes to be affine to the Cauchy process or the generalized Cauchy process is a bit misleading since the latter are well known in the mathematical literature as non-Gaussian Markov processes of the jump-type. The possibility of a physical application for the understanding of the non-Debye dielectric relaxation phenomena is discussed. The predictive power of the presented formalism does not seem to surpass the one based on the combination of continuous time random walks with Lévy stable distributed waiting times, c.f. A. K. Jonscher, A. Jurlewicz and K. Weron [Contemp. Phys. 44, 329 (2003), see also cond-mat/0210481].

Reviewer: Piotr Garbaczewski (Zielona Góra)

MSC:

82B31	Stochastic methods applied to problems in equilibrium statistical mechanics
60G15	Gaussian processes
60G10	Stationary stochastic processes
60G20	Generalized stochastic processes
60K37	Processes in random environments

Keywords:

non-Debye dielectric relaxation process; non-Markovian stationary Gaussian process; Lamperti transformation