Fractional diffusion equations and processes with randomly varying time.

In this paper the solutions u ν = u ν (x, t) to fractional diffusion equations of order 0 < v ≤ 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order v = 1 2 n , n ≥ 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 v = 2 3 n , 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 u ν coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions u ν and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

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