The time-dependent reconstructed evolutionary process with a key-role for mass-extinction events

The homogeneous reconstructed evolutionary process is a birth-death process without observed extinct lineages. Each species evolves independently with the same diversification rates (speciation rate $\lambda(t)$ and extinction rate $\mu(t)$) that may change over time. The process is commonly applied to model species diversification where the data are reconstructed phylogenies, e.g., trees reconstructed from present-day molecular data, and used to infer diversification rates. In the present paper I develop the general probability density of a reconstructed tree under any time-dependent birth-death process. I elaborate on how to adapt this probability density if conditioned on survival of one or two initial lineages, or having sampled $n$ species and show how to transform between the probability density of a reconstructed and the probability density of the speciation times. I demonstrate the use of the general time-dependent probability density functions by deriving the probability density of a reconstructed tree under a birth-death-shift model with explicit mass-extinction events. I enrich this compendium by providing and discussing several special cases, including: the pure birth process, the pure death process, the birth-death process and the critical branching process. Thus, I provide here most of the commonly used birth-death models in a unified framework (e.g., same condition and same data) with common notation.