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Fractional Relaxation Equations and Brownian Crossing Probabilities of a Random Boundary

Part of: Stochastic processes General theory in ordinary differential equations Other special functions

Published online by Cambridge University Press:  04 January 2016

L. Beghin
L. Beghin*
Affiliation:
Sapienza University of Rome
*
∗ Postal address: Department of Statistical Sciences, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Rome, Italy. Email address: [email protected]
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Abstract

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In this paper we analyze different forms of fractional relaxation equations of order ν ∈ (0, 1), and we derive their solutions in both analytical and probabilistic forms. In particular, we show that these solutions can be expressed as random boundary crossing probabilities of various types of stochastic process, which are all related to the Brownian motion B. In the special case ν = ½, the fractional relaxation is shown to coincide with Pr{sup0≤s≤tB(s) < U} for an exponential boundary U. When we generalize the distributions of the random boundary, passing from the exponential to the gamma density, we obtain more and more complicated fractional equations.

Keywords

Fractional relaxation equationgeneralized Mittag-Leffler functionprocesses with random timereflecting and elastic Brownian motioniterated Brownian motionboundary crossing probability

MSC classification

Primary: 60G15: Gaussian processes 34A08: Fractional differential equations 33E12: Mittag-Leffler functions and generalizations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 479 - 505
Copyright
© Applied Probability Trust 

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