Stochastic fractional order differential models on fractals. (English) Zbl 1050.60047

Ruiz-Medina, M. E.; Angulo, J. M.; Ahn, V. V.

Teor. Jmovirn. Mat. Stat. 67, 130-146 (2002); and Theory Probab. Math. Stat. 67, 145-162 (2003).

A class of stochastic fractional diffusion processes represented by fractional order differential operators of elliptic type defined on compact fractal domains is studied. The class of such stochastic models allows the description of fractal random phenomena whose singular behaviour comes from fractal characteristics of the medium, and from the fractional order of regularity of the functional model representing their second-order structure. In the derivation of these models the authors use the theory of generalized random fields, i.e. random distributions defined on fractional Sobolev spaces on fractal domains. Particularly, a class of fractal random fields with reproducing kernel Hilbert space are introduced. To characterize their second-order fractal properties an isomorphic identification between these spaces and suitable fractional Hölder spaces on fractals is established. White noise and fractional-order differential representation on a compact fractal set is derived for elements of the introduced class. In the Gaussian case, the almost sure continuity of sample paths is proved and the corresponding fractional Hölder exponent is calculated as a function of the fractional order of differentiation and of the fractal dimension of the domain.

Reviewer: N. M. Zinchenko (Kyïv)

MSC:

60G20	Generalized stochastic processes
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)

Keywords:

fractional order differential operators of elliptic type; fractal domains; generalized fields; Sobolev space; reproducing kernel Hilbert space