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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.

##### MSC:
 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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