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      Fractional diffusion equations and processes with randomly varying time

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          Abstract

          In this paper the solutions \(u_{\nu}=u_{\nu}(x,t)\) to fractional diffusion equations of order \(0<\nu \leq 2\) are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order \(\nu =\frac{1}{2^n}\), \(n\geq 1,\) we show that the solutions \(u_{{1/2^n}}\) 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 \(\nu =\frac{2}{3^n}\), \(n\geq 1,\) 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 \(u_{\nu}\) coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions \(u_{\nu}\) and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

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          Fractional diffusion and wave equations

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            Fractional Calculus

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              The fundamental solutions for the fractional diffusion-wave equation

              F Mainardi (1996)
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                Author and article information

                Journal
                23 February 2011
                Article
                10.1214/08-AOP401
                1102.4729
                e05924f1-a318-4dc8-8c58-9dea62d04913

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                IMS-AOP-AOP401
                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 Mathematicalitian symmetric spaces of noncompact type. [\FF], \, [\GG] \in H^1(S, \Sigma; \R). Given a measured foliation \FF, we characterize the set of coho Statistics (http://www.imstat.org) We study the symplectic structure of the holomorphic coadjoint orbits, generalizing a theorem of McDuff on the symplectic structure of Hermitian symmetric spaces of noncompact type.
                math.PR
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