What is the conversion rate in a chemical reaction?
The paper you linked to pertains to birth-death processes. An example of a birth-death process in the physical chemistry context that you're interested in  would be a set of molecules where each molecule is in one of two states, $X$ or $Y$, which are isomers of each other. The model assumes that due to the molecules' environment, each isomer can turn into the other isomer at times that are modeled as being random.

In the paper you provided a link to, $w_1$ is the probability per unit time that within a short period of time $\Delta t \ll 1/w_1$, a given molecule that's in state $X$ will transition to state $Y$. I.e., if a given molecule is in state $X$ at time $t$, then there is a probability of $w_1 \Delta t$ that the molecule will have randomly transitioned to state $Y$ by time $t+\Delta t$.