Annals of Probability

Fractional diffusion equations and processes with randomly varying time

Enzo Orsingher and Luisa Beghin

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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.

Article information

Ann. Probab., Volume 37, Number 1 (2009), 206-249.

First available in Project Euclid: 17 February 2009

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Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory 60G52: Stable processes 60J65: Brownian motion [See also 58J65]
Secondary: 33E12: Mittag-Leffler functions and generalizations 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$

Iterated Brownian motion fractional derivatives Airy functions McKean law Gauss–Laplace random variable stable distributions


Orsingher, Enzo; Beghin, Luisa. Fractional diffusion equations and processes with randomly varying time. Ann. Probab. 37 (2009), no. 1, 206--249. doi:10.1214/08-AOP401.

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