Selfsimilarity of diffusions’ first passage times

Abstract

Considering a general diffusion process that runs over the non-negative half-line, this paper addresses the first-passage time (FPT) to the origin: the time it takes the process to get from an arbitrary fixed positive level to the level zero. Inspired by the special features of Brownian motion, three types of FPT selfsimilarity are introduced: (i) stochastic, which holds in ‘real space’; (ii) Laplace, which holds in ‘Laplace space’; and (iii) joint, which is the combination of the stochastic and Laplace types. Analysis establishes that the three types of FPT selfsimilarity yield, respectively and universally, the following FPT distributions: inverse-gamma; inverse-Gauss; and Levy–Smirnov. Moreover, the analysis explicitly pinpoints the classes of diffusion processes that produce the three types of selfsimilar FPTs. Shifting from general diffusion dynamics to Langevin dynamics, it is shown that the three classes collapse, respectively, to the following specific processes: diffusion in a logarithmic potential; Brownian motion with drift; and Brownian motion. Also, the effect of the Girsanov transformation on the three types of selfsimilar FPTs is investigated, as well as the effect of initiating the diffusion process from its steady-state level (rather than from a fixed positive level). This paper presents a novel approach to the exploration of first-passage times.