A partial differential equation (also called the forward Kolmogorov equation) that governs the time evolution of the probability density function of a Markov diffusion process by encoding drift and diffusion (noise) terms.

Principle

It embodies conservation of probability under the action of a generator: advection of density by deterministic drift and spreading by diffusion; equivalently, it is the ensemble-level representation of stochastic differential equations (Langevin form).

Demonstration

Explicit solution for the Ornstein–Uhlenbeck process is Gaussian with analytically evolving mean and variance; gradient drift with quadratic potential yields equilibrium Maxwell–Boltzmann type densities illustrating relaxation to steady state.

Misapplication

Using the classical Fokker–Planck PDE for processes with jumps, heavy-tailed Lévy statistics, or pronounced non-Markovian memory without replacing diffusion by appropriate nonlocal or fractional operators produces incorrect dynamics.

Consequence

When valid, the Fokker–Planck equation yields time-dependent densities, stationary distributions, spectral information about relaxation rates, and a pathway to compute moments, fluxes, and large-deviation properties.

Reversal

The backward Kolmogorov (backward equation) focuses on evolution of transition expectations rather than densities; Liouville's equation is the deterministic limit (zero-noise) counterpart describing phase-space transport without diffusion.

Boundary

Applies to continuous-path Markov processes with sufficiently smooth drift and diffusion coefficients and well-posed boundary conditions; excludes jump processes, purely discrete-state Markov chains, and strongly non-Markovian dynamics unless extended appropriately.

Semantic Tension

Tension between the Fokker–Planck (PDE, ensemble) viewpoint and the Langevin (trajectory) viewpoint: both are equivalent under conditions but suggest different approximation strategies; tension also with master equations for jump processes.

Synthesis

The Fokker–Planck equation translates stochastic differential dynamics into a deterministic PDE for probability densities, encoding drift and diffusion so one can analyze transient and steady probabilistic behavior, spectral relaxation, and macroscopic fluxes.