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Mellin transform and subordination laws in fractional diffusion processes. (English) Zbl 1083.60032
The article outlines the role of Mellin transforms for fractional diffusion processes. For suitable functions $f$ its Mellin transform is defined as ${f}^{*}\left(s\right)={\int }_{0}^{\infty }f\left(x\right)\phantom{\rule{0.166667em}{0ex}}{x}^{s-1}\phantom{\rule{0.166667em}{0ex}}dx,\phantom{\rule{0.166667em}{0ex}}s\in ℂ,$ whenever the integral exists. A convolution formula for Mellin transforms is known to correspond to the probability density function of the product of two independent nonnegative random variables. The authors’ main point is the usefulness of Mellin transforms for the derivation of fractional diffusion processes governed by generalized diffusion equations with fractional order derivatives in space and/or in time. The natural way to obtain these subordinated selfsimilar stochastic processes is to consider them as limiting cases of appropriate continuous time random walk models. The authors elaborate on how particular subordination formulas can be found by purely analytic methods by the use of Mellin transforms for the derivation of fundamental solutions of fractional diffusion equations, allowing in turn a stochastic interpretation of the formulas.
##### MSC:
 60G18 Self-similar processes 26A33 Fractional derivatives and integrals (real functions) 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 44A15 Special transforms (Legendre, Hilbert, etc.) 44A35 Convolution (integral transforms) 60G52 Stable processes
##### Keywords:
Mellin-Barnes integral; selfsimilar process