Diffusion Approximation

Diffusion approximation is the mathematical technique, developed by Sewall Wright and later formalized by Motoo Kimura, that treats allele frequency dynamics in a population as a continuous stochastic diffusion process rather than a discrete Wright-Fisher Markov chain. The approximation replaces the exact binomial sampling of the Wright-Fisher model with a stochastic differential equation driven by Brownian motion, scaled by the variance effective population size.

The power of the diffusion approximation lies in its analytical tractability. Whereas exact Wright-Fisher dynamics require combinatorial calculations, the diffusion approach yields partial differential equations — the Kolmogorov backward and forward equations — that can be solved for quantities like fixation probabilities, mean absorption times, and stationary distributions under selection and mutation. These solutions are approximations, but they are often remarkably accurate even for moderate population sizes.

The technique extends beyond genetics. Any system with discrete states, small step sizes, and demographic stochasticity can be approximated by diffusion — from random walks in ecology to stochastic processes in financial mathematics. The approximation fails, however, when populations are small or when selection is strong enough to produce discontinuous jumps. In these regimes, the continuous assumption breaks down and the exact discrete model must be used.