Fokker–Planck equations and probability transport

For a time-homogeneous Itô diffusion dX=b(X)dt+sigma(X)dB, apply Itô's formula to a smooth compactly supported test function f. Under suitable integrability, the stochastic integral has mean zero. The expectation of f therefore changes according to the generator Lf=b f′+(sigma²/2)f″. This weak formulation remains meaningful in settings where a classical smooth density has not yet been established. Existence of a density and enough regularity to differentiate it are separate analytical questions, not consequences of writing the SDE symbolically.

When a sufficiently smooth density p exists, write the expectation as the integral of f against p. Move derivatives from f to the coefficients times p using integration by parts. The first derivative gains a minus sign; the second derivative retains a plus sign. Boundary terms vanish for compactly supported probes, while global probability conservation still requires appropriate boundary or tail conditions. The resulting forward operator is the adjoint L* acting on distributions or densities, rather than the generator L acting on observables.

Define probability flux J=bp−one-half the spatial derivative of sigma squared times p. Then the forward equation says that density decreases where more probability leaves than enters. Integrating across an interval gives the difference between incoming and outgoing boundary fluxes. Reflecting boundaries impose zero flux. Absorbing boundaries permit loss of probability from the interior; tracking that lost mass as absorption restores a complete probabilistic description. The boundary rule is part of the model and cannot be recovered from the interior differential equation alone.

For constant noise sigma=√(2D), pure Brownian diffusion has equation pₜ=D pₓₓ. Its fundamental Gaussian solution has variance 2Dt, which checks the coefficient. Confusing sigma with D introduces a factor-of-two error. For state-dependent noise, the diffusion term is one-half the second derivative of sigma²p, not merely one-half sigma² times pₓₓ. Derivatives of the coefficient matter. A change from Itô to Stratonovich notation also changes the corresponding drift unless the appropriate conversion is made.

For the Ornstein–Uhlenbeck process dX=−kappa X dt+√(2D)dB, with positive kappa and D, the restoring drift pulls toward zero while diffusion spreads probability. On the real line, a normalizable zero-flux stationary density solves D p′=−kappa x p. Integrating the logarithmic derivative produces a Gaussian with variance D/kappa. The negative exponent is essential: changing its sign would yield a nonintegrable function that grows in the tails rather than a probability density.

Starting from a deterministic x₀, the mean decays as x₀e^(−kappa t), while variance grows from zero toward D/kappa. This separates relaxation of the center from spreading around it. Stationarity means the full distribution remains unchanged, not that sample paths stop fluctuating. Zero flux characterizes this equilibrium on the line with natural decay conditions, but it is not a universal requirement for every stationary system: a process on a circle can carry a nonzero constant circulating current while retaining a stationary density.