Fractional relaxation equations and Brownian crossing probabilities of a random boundary

Abstract

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{sup_0≤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.