Wright–Fisher diffusion ($d{X}_t=\sqrt{X_t(1-X_t)}d{B}_t$)

The neutral one-dimensional Wright–Fisher diffusion records a proportion in $[0,1]$, with absorbing endpoints, zero drift and generator $Lf(x)=x(1-x)f^{\prime \prime }(x)/2$. Its conditional variance rate is largest near one-half and vanishes at fixation. Mutation or selection variants add drift and must be specified separately.

An urn with $j$ red balls among $n$ is sampled independently $n$ times with replacement, and the next generation's red count is the number of red draws. Its proportion is a bounded discrete-time martingale. Accelerating generation time by $n$ gives a Wright–Fisher diffusion limit: the one-step variance of the proportion is $x(1-x)/n$, and the higher moments make Taylor remainders negligible.