PROCEEDINGS OF THE RUSSIAN HIGHER SCHOOL
ACADEMY OF SCIENCES

Abstract
A large amount of experimental data on various processes of the so-called anomalous diffusion for which variance varies nonlinearly over time has currently been accumulated. Various methods of modeling anomalous diffusion are associated with such properties of the corresponding processes as “a strong form” of increment dependence and increment nonstationarity (see, e.g., [1]–[4].). The well-known examples of such processes are continuous time random walk (CTRW) models and the fractional Brownian motion (see e. g, [4], [5]). Today, apparently, there are no modeling formats (see. [3]) covering all of these properties, similar to the Wiener process which is a classical format of the Brownian motion. Questions of modeling transport processes in singular phase spaces were raised in [1]–[4] etc., where the modeling of transport processes in continuous media with a fractal structure was studied. These processes were considered as a subset of the zero Lebesgue and some non-zero Hausdorff measures. A technique of fractional integro-differential calculus was used as a modeling tool in these studies. In this paper we depart from the paradigm that transfer processes are modeled in continuous media with a fractal structure. We construct a master equation that makes it possible to model anomalous diffusion processes in such a way as to take into account both the aftereffect fractal structure and correlation properties of the process. This master equation allows obtaining the Wiener process and the fractional Brownian motion as limiting cases. This present paper is a natural continuation of a series of papers [6], [7], [8], [9] in which an anomalous character of mass, energy, and momentum transport was closely linked with the introduction of values singular relative to the Lebesgue measure.

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