the Fokker-Planck equation
/ FOK-er PLANK /
The Fokker-Planck equation (also called the Kolmogorov forward equation) is the partial differential equation governing how the probability density of a diffusion evolves in time. If a diffusion X_t solves dX = b(X) dt + sigma(X) dB and has a density p(t, x) for its position at time t, then p satisfies a parabolic PDE driven by the formal adjoint of the diffusion generator. It answers the question: starting from a known initial distribution, how does the cloud of probability mass spread, drift, and (possibly) settle into a steady state?
The equation is partial_t p(t, x) = L^* p(t, x), where L^* is the formal adjoint of the generator L f = sum_i b_i partial_i f + (1/2) sum_{ij} a_ij partial_i partial_j f. Explicitly, L^* p = - sum_i partial_i (b_i(x) p) + (1/2) sum_{ij} partial_i partial_j (a_ij(x) p), with a = sigma sigma^T. The first term is a drift/advection term (it transports mass along the drift field b) and the second is a diffusion term (it smears mass out). Contrast this with the backward Kolmogorov equation partial_t u = L u, which evolves an observable E[f(X_t) given X_0 = x] in the starting point x; the forward equation evolves the density in the end point x. The same generator runs both, one through L and one through its adjoint L^*. Stationary (invariant) densities are the time-independent solutions, characterized by L^* p = 0, which in one dimension can often be solved explicitly to give p proportional to exp(-2 integral b/a) / a (the detailed-balance form).
Fokker-Planck is the central tool of nonequilibrium statistical physics and of any study of how distributions relax: it tells you the transient spreading, the approach to equilibrium, and the stationary law. Honest cautions: the equation is for the DENSITY, which must exist — it does when the diffusion is nondegenerate (uniformly elliptic) and smooth, but a degenerate diffusion may have no density (mass concentrated on a sub-manifold), and then Fokker-Planck must be read in the weak/measure sense. Also, conservation of total probability (integral p dx = 1) holds only if there is no explosion and no flux loss at the boundary; on a bounded domain the boundary conditions (reflecting, absorbing) are part of the problem and change the answer.
For the OU process dX = -theta X dt + sigma dB, Fokker-Planck reads partial_t p = theta partial_x(x p) + (sigma^2/2) partial_xx p. The stationary solution L^* p = 0 gives p(x) proportional to exp(-theta x^2 / sigma^2), i.e. the Gaussian N(0, sigma^2/(2 theta)) — the diffusion's equilibrium distribution.
Fokker-Planck for the OU process recovers its Gaussian equilibrium as the steady state L* p = 0.
Fokker-Planck governs the density, which need not exist — a degenerate (non-elliptic) diffusion may put mass on a lower-dimensional set, and then the equation holds only weakly. The forward equation uses the adjoint L*, the backward equation uses L itself; do not confuse them.