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.

考虑到在非负半线上运行的一般扩散过程，本文讨论了到达原点的首次通过时间（FPT）：从任意固定的正电平到零电平所花费的时间。受布朗运动特性的启发，引入了三种类型的FPT自相似性：（i）随机，在“真实空间”中保持；（ii）拉普拉斯，位于“拉普拉斯空间”内；（iii）联合，即随机和拉普拉斯类型的组合。分析确定三种类型的FPT自相似性分别普遍产生以下FPT分布：反伽马；反高斯 和利维·史密诺夫（Levy–Smirnov）。此外，分析明确指出了产生三种自相似FPT的扩散过程的类别。从一般扩散动力学转变为兰格文动力学，结果表明这三个类别分别崩溃为以下特定过程：对数势中的扩散；带漂移的布朗运动；和布朗运动。此外，研究了吉萨诺夫变换对三种自相似FPT的影响，以及从其稳态水平（而不是从固定的正水平）启动扩散过程的影响。本文提出了一种探索首次通过时间的新颖方法。研究了Girsanov变换对三种自相似FPT的影响，以及从其稳态水平（而不是从固定的正水平）启动扩散过程的影响。本文提出了一种探索首次通过时间的新颖方法。研究了Girsanov变换对三种自相似FPT的影响，以及从其稳态水平（而不是从固定的正水平）启动扩散过程的影响。本文提出了一种探索首次通过时间的新颖方法。