= Ornstein-Uhlenbeck fluctuation limit at a stable population equilibrium
{c}
{title2=$dX_t=-\alpha X_tdt+\alpha\sqrt{2/\beta}\,dB_t$}
For a unit-jump population chain with one-step probabilities $\alpha k/n$ and $\beta k^2/n^2$, its equilibrium density is $u_0=\alpha/\beta$. Speed time by $n$ and center and scale the population as $(k-nu_0)/\sqrt n$. The accelerated conditional drift is $-\alpha x-\beta x^2/\sqrt n$, and the second moment is $2\alpha^2/\beta+3\alpha x/\sqrt n+\beta x^2/n$. Their compact-set limits give the displayed <Ornstein-Uhlenbeck process>. If $2\alpha^2<\beta$, the invalid-probability threshold stays a fixed positive density distance above equilibrium. Before that threshold, the generator of $x^2$ is at most one because its drift contribution is nonpositive. The stopped <Lyapunov function> bound proves <compact containment> and makes the probability of hitting that threshold on a fixed accelerated horizon tend to zero. This supplies the global hypotheses of the <diffusion approximation theorem>.
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