Fractional relaxation equations and Brownian crossing probabilities of a random boundary

TL;DR

This paper explores fractional relaxation equations of order ν in (0,1), deriving solutions in analytical and probabilistic forms, linking them to crossing probabilities of Brownian motion with random boundaries, including exponential and Gamma distributions.

Contribution

It introduces a novel probabilistic interpretation of fractional relaxation equations as crossing probabilities of Brownian motion with various random boundaries.

Findings

Solutions expressed as crossing probabilities of Brownian motion.

Special case ν=1/2 relates to exponential boundary crossing.

Generalization to Gamma boundaries leads to complex fractional equations.

Abstract

We analyze here different forms of fractional relaxation equations of order {\nu}\in(0,1) and we derive their solutions both in analytical and in probabilistic forms. In particular we show that these solutions can be expressed as crossing probabilities of random boundaries by various types of stochastic processes, which are all related to the Brownian motion B. In the special case {\nu}=1/2, the fractional relaxation is proved to coincide with Pr{sup_{0\leqs\leqt} B(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.