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Fractional diffusion equations and processes with randomly varying time. (English) Zbl 1173.60027
The authors consider time-fractional diffusion equations of order \(0<\nu\leq2\) and interprete their solutions as densities of the composition of various types of stochastic processes. It is proved that the solution for \(\nu=1/2^n\) corresponds to the distribution of \(n\)-times iterated Brownian motion. The explicit solutions of the fractional diffusion equation are presented for \(\nu=1/3, 2/3, 4/3.\) In particular, \(u_{2/3}(x,t)\) is given in terms of Airy functions. The relationship between the solution \(u_{\nu}(x,t)\) and the stable densities is presented. The behavior of the solution (for \(x\) varying and \(t\) fixed), which is substantially different in the two intervals \(0<\nu\leq1\) and \(1<\nu\leq2\) is analyzed.

60J60 Diffusion processes
26A33 Fractional derivatives and integrals
60G52 Stable stochastic processes
60J65 Brownian motion
33E12 Mittag-Leffler functions and generalizations
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
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