In this paper the solutions uν=uν(x, t) to fractional diffusion equations of order 0<ν≤2 are analyzed and interpreted as densities of the composition of various types of stochastic processes.
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For the fractional equations of order ν=1/2n, n≥1, we show that the solutions u1/2n correspond to the distribution of the n-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order ν=2/3n, n≥1, is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions.
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In the general case we show that uν coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions uν and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.
@article{1234881689,
author = {Orsingher, Enzo and Beghin, Luisa},
title = {Fractional diffusion equations and processes with randomly varying time},
journal = {Ann. Probab.},
volume = {37},
number = {1},
year = {2009},
pages = { 206-249},
language = {en},
url = {http://dml.mathdoc.fr/item/1234881689}
}
Orsingher, Enzo; Beghin, Luisa. Fractional diffusion equations and processes with randomly varying time. Ann. Probab., Tome 37 (2009) no. 1, pp. 206-249. http://gdmltest.u-ga.fr/item/1234881689/