Constant-birth pair-annihilation process

A count-valued Markov jump process with constant single-particle births and pair annihilation must preserve $n\in {\mathbb{Z}}_{\ge 0}$. Choose the ratesThe second rate vanishes at $n=0,1$, where there is no pair to remove; the factor $1/2$ for unordered pairs has been absorbed into $\beta$.

In the Kramers-Moyal expansion, the first two coefficients are the rate-weighted first and second jump moments:Here $B$ is the noise variance per unit time in the local diffusion approximation. The positive drift root isThis is the stable fixed point of the drift equation, rather than an exact formula for the mean of the stochastic population.

For $n_{\ast }\gg 1$, fluctuations far from the count boundary are approximately an Ornstein-Uhlenbeck process. Its stationary variance isAt leading large-count order one may replace $n(n-1)$ by $n^2$ and $n_{\ast }$ by $\sqrt{\lambda /(2\beta )}$, recovering the simpler approximation. A rate proportional to $n^2$ used literally at $n=1$ would allow a negative population, so that replacement is an interior asymptotic approximation rather than the exact count model.