2009/01/01 by Enzo Orsingher, Luisa Beghin · 4 citations
Economics, Econometrics and Finance · Mathematics · #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Stochastic processes and financial applications #math.PR
paper · pdf · doi:10.1214/08-aop401
published as Annals of Probability 2009, Vol. 37, No. 1, 206-249 · Published in at http://dx.doi.org/10.1214/08-AOP401 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2009/01/01 · arxiv created 2011/02/23 · arxiv updated 2011/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.