Fractional relaxation equations and Brownian crossing probabilities of a random boundary (Q2898916)

The relaxation equation is well-known in the physical literature, in particular, in the study of the rheological properties of materials. Its fractional version (obtained by substituting the time-derivative with a fractional one) has also been presented in connection with the electromagnetic properties of a wide range of materials (which exhibit long memory, instead of exponential decay).NEWLINENEWLINEThis paper analyzes different fractional extensions of the relaxation equation, whose solutions can be represented, in probabilistic terms, either as survival probabilities of fractional renewal processes, or as random boundary crossing probability of various stochastic processes.NEWLINENEWLINEIn particular, the fractional relaxation-type equations examined here are proved to govern the crossing probability of a random boundary (with uniform or gamma distribution) by some processes all linked to the Brownian motion; the first passage time of a Brownian motion by a level \(t\), the Bessel process, the elastic Brownian motion. Only when the sojourn time of a Brownian motion on the positive half-axis is considered a non-fractional, second-order equation is obtained.