the backward and forward Kolmogorov equations
/ kol-mo-GO-rov /
The backward and forward Kolmogorov equations are the two partial differential equations governing a Markov transition function, obtained by differentiating the semigroup law P_(t+s) = P_t P_s in two different ways. They are the analytic incarnation of the process: one evolves observables, the other evolves probability densities, and together they are how diffusion theory connects probability to PDEs.
Backward equation: differentiate P_(t+s) = P_s P_t at s = 0 acting from the LEFT to get d/dt P_t = A P_t, i.e. for u(t, x) = (P_t f)(x) = E_x[ f(X_t) ], the function solves du/dt = A u with u(0, .) = f, where A acts on the BACKWARD (starting-point) variable x. This is the equation for expectations and is the one underlying Feynman-Kac. Forward equation: differentiate from the RIGHT to get d/dt P_t = P_t A; in terms of the density, if X_t has density p(t, y) (or the transition density p_t(x, y) as a function of the endpoint y), it solves dp/dt = A^* p, where A^* is the FORMAL ADJOINT of the generator acting on the forward (endpoint) variable. For a diffusion with generator A f = (1/2) sum a_(ij) d_i d_j f + sum b_i d_i f, the adjoint gives the Fokker-Planck equation dp/dt = (1/2) sum d_i d_j (a_(ij) p) - sum d_i (b_i p) — a conservation law transporting probability mass.
Why it matters: the two equations let you compute either expectations (backward) or distributions (forward) by solving a PDE, and their stationary forms give harmonic functions (A u = 0) and invariant densities (A^* p = 0). Honest cautions worth keeping: the forward equation requires a density to exist and enough smoothness of the coefficients to take the adjoint derivatives, while the backward equation needs f in the domain — they are not always simultaneously valid; the boundary conditions differ (the backward variable carries the domain's boundary behaviour, the forward variable carries mass-flux conditions); and 'A^* is the adjoint' is with respect to the Lebesgue (or reference) measure, so the precise form of Fokker-Planck depends on that choice — under the invariant measure the picture simplifies dramatically (reversibility makes A self-adjoint).
For Brownian motion A = (1/2) d^2/dx^2 is self-adjoint, so backward and forward equations coincide as the heat equation dp/dt = (1/2) d^2 p/dx^2. For the Ornstein-Uhlenbeck generator A f = -x f' + f'' the adjoint differs: A^* p = (xp)' + p'', whose stationary solution A^* p = 0 is the Gaussian density p proportional to e^(-x^2/2), the invariant measure.
Backward evolves expectations by A; forward (Fokker-Planck) evolves densities by the adjoint A^*; for self-adjoint A they coincide.
The forward (Fokker-Planck) equation needs a density and smooth enough coefficients to form the adjoint; backward needs f in the domain. The two are not interchangeable, and their boundary conditions differ.