Birth-death processes are time-changed Feller’s Brownian motions

A Feller’s Brownian motion refers to a Feller process on the interval $[0,\infty )$ that is equivalent to the killed Brownian motion before reaching 0. It is fully determined by four parameters $(p_1,p_2,p_3,p_4)$, reflecting its killing, reflecting, sojourn, and jumping behaviors at the boundary 0. On the other hand, a birth–death process is a continuous-time Markov chain on $N$ with a given birth–death $Q$-matrix, and it is characterized by three parameters $(\gamma ,\beta ,\nu )$ that describe its killing, reflecting, and jumping behaviors at the boundary $\infty$. The primary objective of this paper is to establish a connection between Feller’s Brownian motion and birth–death process. We will demonstrate that any Feller’s Brownian motion can be transformed into a specific birth–death process through a unique time change transformation, and conversely, any birth–death process can be derived from Feller’s Brownian motion via time change. Specifically, the birth–death process generated by the Feller’s Brownian motion, determined by the parameters $(p_1,p_2,p_3,p_4)$, through time change, has the parameters: $\gamma =p_1,\beta =2p_2,{\nu }_n=p_n,n\in N,$where $\{p_n:n\in N\}$ is a sequence derived by allocating weights to the measure $p_4$ in a specific manner. Utilizing the pathwise representation of Feller’s Brownian motion, our results provide a pathwise construction scheme for birth–death processes, addressing a gap in the existing literature.