Fractional diffusion equations and processes with randomly varying time.

By Enzo Orsingher, Luisa Beghin. 2011

137 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/53ac387e0b7e2bbb59bc47cc6f0e24c5d5bb19aa

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Citations in the corpus

Poisson-type processes governed by fractional and higher-order recursive differential equations (2009, background): “1 Introduction Many well-known differential equations have been extended by introducing fractional-order derivatives with respect to time (for instance, the heat and wave equations ([14]-[15] and [21]) as well as the telegraph equation ([20]) and the higher-order heat-type equations ([1])) or with r”
On a fractional linear birth-death process (2011, unknown): “12) × ∫ ∞ 0 e 2 n−1/(4t) √ 4πt dωn−1, as can easily be inferred from [7], Theorem 2.”
Semilinear Caputo time-fractional pseudo-parabolic equations (2021, background): “Many important physical models and practical problems require us to consider the pseudo-parabolic model with fractional derivative rather than classical one, like the physical model considering memory effects [18, 19, 13, 48] and some corresponding engineering problems [4, 5, 7, 44, 46, 50, 56], esp”
Fractional Poisson processes and related planar random motions (2009, methodology): “We can analogously consider the planar random motion changing direction at fractional Poisson times of order ν = 1 4 , by applying the results obtained in Orsingher and Beghin (2009).”
Convolution-Type Derivatives, Hitting-Times of Subordinators and Time-Changed C0-semigroups (2013, background): “[27] E. Orsingher and L. Beghin.”
Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives ✩ (2012, background): “Orsingher and Beghin [10] analyzed the solutions to fractional diffusion equations of order α ∈ (0,2], and interpreted them as densities of the composition of various types of stochastic processes.”