What does this equation mean?

Motoo Kimura, at Japan’s National Institute of Genetics, made the borrowing exact rather than approximate. His 1955 paper in the Proceedings of the National Academy of Sciences solved the full time-dependent diffusion equation derived from the Wright-Fisher model, recovering not just the eventual resting distribution of an allele’s frequency but its whole transient shape as it evolves generation by generation [ 6 ] . His 1962 paper in Genetics then posed the general fixation problem as a diffusion boundary-value equation and named the lineage precisely: the question of a mutant gene’s ultimate fate was “first treated quantitatively by Fisher (1922),” with “equivalent results” reached independently by Haldane in 1927 and Wright in 1931 [ 7 ] . Kimura wrote the fixation probability u(p,t) as the solution of what he explicitly called the Kolmogorov backward equation, subject to the boundary conditions u(0,t)=0 and u(1,t)=1 — an allele already lost stays lost, one already fixed stays fixed [ 7 ] . That boundary structure is doing real work: it is the diffusion equation’s way of encoding that zero and one are absorbing states of a finite population, not just convenient endpoints on a graph. Set the two founding equations beside each other and the claim of one shared mathematics stops being a figure of speech: $\frac{\partial \phi(p,t)}{\partial t} = \frac{1}{2}\,\frac{\partial^{2}}{\partial p^{2}}\Big[V(p)\,\phi(p,t)\Big], \quad V(p) = \frac{p(1-p)}{2N} \qquad \text{(Kimura, 1955/1962)}$. The same second-derivative operator acts on a diffusion term in both; only the variable and the coefficient it multiplies have changed. One honest difference is worth stating rather than smoothing over: Einstein’s D is a constant, fixed once by temperature and viscosity, while Kimura’s V(p) depends on the frequency itself, so the spread is fastest near p=0.5 and vanishes at the boundaries — an allele already fixed or already lost has nothing left to drift into [ 7 ] . The operator is identical; the coefficient it acts on has learned something about the biology.