Fokker–Planck equation

Definition

Let $n, d \ge 1$. Let $\mu \colon \R^n \times [0, \infty) \to \R^n$ and $\sigma \colon \R^n \times [0, \infty) \to \R^{n \times d}$ be smooth. Let $D = \frac{1}{2}\sigma \sigma^T$, a symmetric positive semidefinite $n \times n$ matrix at each point. The Fokker–Planck equation with drift $\mu$ and diffusion matrix $D$ is the equation for $p(x, t)$: $$\partial_t p = -\sum_{i=1}^{n} \partial_{x_i}\big(\mu_i\, p\big) + \sum_{i,j=1}^{n} \partial_{x_i} \partial_{x_j}\big(D_{ij}\, p\big),$$ where $\partial$ denotes partial derivatives. It is associated with the stochastic differential equation $dX_t = \mu(X_t, t)\, dt + \sigma(X_t, t)\, dW_t$, where $W$ is a $d$-dimensional Wiener process.