A parabolic partial differential equation that describes the time evolution of the probability density function of the state of a stochastic process, typically the diffusion limit of a stochastic differential equation's law.

Principle

Express conservation of probability and flux balance: drift terms create advective flux, diffusion terms create diffusive flux; together they govern density transport and approach to invariant measures.

Demonstration

The Fokker–Planck equation associated with the Ornstein–Uhlenbeck process is a linear parabolic PDE whose Gaussian fundamental solution relaxes to a stationary Gaussian density when the drift is restoring.

Misapplication

Applying Fokker–Planck formulas derived for continuous diffusions to jump processes or discrete-state Markov chains without replacing with a master equation or nonlocal operator.

Consequence

Transforms a stochastic dynamics problem into a deterministic PDE for densities: steady solutions correspond to invariant measures, spectral analysis of the generator yields rates of relaxation and variance decay.

Reversal

Removing diffusion terms yields the Liouville transport equation for deterministic flows (pure advective transport); adding nonlocal jump terms converts it to an integrodifferential master-type equation.

Boundary

Valid for Markov diffusion processes with sufficiently regular coefficients and appropriate boundary conditions; does not directly apply to pure jump processes or ill-posed coefficient regimes requiring distributional solutions.

Semantic Tension

Often compared with the backward Kolmogorov equation: the Fokker–Planck is forward in time for densities, while the backward equation governs observables or expectations evolving backward along the generator.

Synthesis

A deterministic PDE encoding the forward-in-time evolution of probability densities for diffusion-type stochastic processes, linking drift and diffusion coefficients to advective and diffusive probability fluxes and steady states.