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January 2009 Fractional diffusion equations and processes with randomly varying time
Enzo Orsingher, Luisa Beghin
Ann. Probab. 37(1): 206-249 (January 2009). DOI: 10.1214/08-AOP401


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.

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.

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.


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Enzo Orsingher. Luisa Beghin. "Fractional diffusion equations and processes with randomly varying time." Ann. Probab. 37 (1) 206 - 249, January 2009.


Published: January 2009
First available in Project Euclid: 17 February 2009

zbMATH: 1173.60027
MathSciNet: MR2489164
Digital Object Identifier: 10.1214/08-AOP401

Primary: 60E05 , 60G52 , 60J65
Secondary: 33C10 , 33E12

Keywords: Airy functions , fractional derivatives , Gauss–Laplace random variable , iterated Brownian motion , McKean law , Stable distributions

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 1 • January 2009
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