 Fokker–Planck equation

A Fokker-Planck equation describes how a probability density evolves under drift and diffusion. For an Itô diffusion $d{X}_t=b(X_t,t)dt+\sigma (X_t,t)d{W}_t$, put $a=\sigma {\sigma }^{\mathsf{T}}$. When a density exists and the required regularity holds, its equation isThe first term transports probability with the drift; the second spreads it through the local increment covariance. It is the forward equation adjoint to the backward infinitesimal generator $Lf=b\cdot \nabla f+\frac{1}{2}{\sum }_{i,j}{a}_{ij}{\partial }_i{\partial }_{j}f$.

Writing $J_i=b_{i}p-\frac{1}{2}{\sum }_j{\partial }_j(a_{ij}p)$ gives the continuity equation ${\partial }_{t}p+\nabla \cdot J=0$. Boundary conditions specify whether probability is reflected, absorbed or exchanged with the exterior. An invariant density must satisfy both the stationary equation and the boundary conditions. A zero divergence of current need not mean zero current.

For constant drift zero and $a=2DI$, the equation reduces to the heat equation ${\partial }_{t}p=D\Delta p$; a point source spreads into a Normal distribution with covariance $2DtI$. For one-dimensional drift $b(x)=-\gamma x$ with $\gamma >0$ and constant $a=2D>0$, the stationary density is normal with variance $D/\gamma$. Some texts call $a$ the diffusion matrix, while others use $a/2$; stating the displayed convention avoids a factor-of-two ambiguity.

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.