A probability density left unchanged by the Fokker-Planck equation . Equivalently its Fokker-Planck probability current has zero divergence; the current itself need not vanish. For the Ornstein-Uhlenbeck Fokker-Planck equation, the normalized standard Gaussian density has zero current and is stationary.
If and , then . Insert compact cutoffs , integrate the backward equation in time, and integrate by parts twice to move spatial derivatives onto . The error is bounded by . When these norms are finite it vanishes, proving without assuming derivative behavior at infinity.
The probability-density equation for the Ornstein-Uhlenbeck process with drift and diffusion coefficient . The unit-covariance Gaussian density is stationary because its Fokker-Planck probability current vanishes. This normalization makes mean velocity decay at rate one and energy excess decay at rate two.
For a sufficiently regular unit-mass solution, the Ornstein-Uhlenbeck Fokker-Planck equation is . Integration by parts proves the displayed identity. Consequently relative entropy decreases, and its dissipation is relative Fisher information.
The inequality and the entropy dissipation identity for Ornstein-Uhlenbeck flow imply . An integrating factor proves the displayed estimate from initial entropy alone. Pinsker's inequality then gives .
If finite nonnegative entropy decreases, its limit exists. The fundamental theorem of calculus applied to proves the displayed absolute-integrability identity. This product conclusion does not require a zero limiting entropy.
Integration by parts against , and gives these equations for mass, unnormalized momentum and kinetic energy. With , the mean is , and . Normalizing by mass is invalid if that mass is zero.
The diffusion matrix is the infinitesimal second jump moment, . It is symmetric and positive semidefinite and multiplies the second-derivative term of the Fokker-Planck equation.
The drift vector is the expected infinitesimal displacement per unit time. For jumps with rates , it is . It is the first jump moment entering the Fokker-Planck equation.
For linear drift and constant symmetric positive-definite diffusion matrix , a zero-current stationary Gaussian obeysThis reversible solution exists when is symmetric positive definite. Its covariance satisfies the Continuous Lyapunov equation
A reflecting boundary confines a diffusion without losing probability. Its forward condition is zero outward Fokker-Planck probability current, , and its backward condition is a zero conormal derivative.
A zero-flux boundary condition sets the normal probability current to zero. It conserves total probability inside the domain and represents reflection for a confined diffusion.
An absorbing boundary removes a diffusion when it is reached. The forward density and backward observables satisfy Dirichlet boundary conditions there.
A partially absorbing boundary removes a particle at finite surface reactivity and reflects it otherwise. Its forward Robin boundary condition is . If the normal diffusion coefficient is , integration by parts gives the backward condition
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The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of forces, such as random fluctuations or deterministic forces. It is commonly used in various fields, including statistical mechanics, diffusion processes, and financial mathematics, to model systems that exhibit stochastic behavior.