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  "body": "\n## Definition\n\nLet $n, d \\ge 1$.\nLet $\\mu \\colon \\R^n \\times [0, \\infty) \\to \\R^n$ and $\\sigma \\colon \\R^n \\times [0, \\infty) \\to \\R^{n \\times d}$ be smooth.\nLet $D = \\frac{1}{2}\\sigma \\sigma^T$, a symmetric positive semidefinite $n \\times n$ matrix at each point.\nThe *Fokker–Planck equation* with drift $\\mu$ and diffusion matrix $D$ is the equation for $p(x, t)$:\n$$\\partial_t p = -\\sum_{i=1}^{n} \\partial_{x_i}\\big(\\mu_i\\, p\\big) + \\sum_{i,j=1}^{n} \\partial_{x_i} \\partial_{x_j}\\big(D_{ij}\\, p\\big),$$\nwhere $\\partial$ denotes [[partial derivatives|09EGC6F8PT]].\nIt is associated with the [[stochastic differential equation|B219G9SHJA]] $dX_t = \\mu(X_t, t)\\, dt + \\sigma(X_t, t)\\, dW_t$, where $W$ is a $d$-dimensional [[Wiener process|22735FGJV6]].\n\n## Remarks\n\nUnder suitable regularity of $\\mu$ and $\\sigma$, if $X_t$ has a smooth [[probability density|0ZY9DX8FBF]] $p(\\cdot, t)$, then $p$ solves this equation.\nExample: for $n = d = 1$, $\\mu = 0$ and $\\sigma = 1$, the equation is $p_t = \\frac{1}{2} p_{xx}$.\nIts solution $p(x, t) = (2\\pi t)^{-1/2} e^{-x^2/(2t)}$ is the density of $W_t$; one checks $p_t = \\frac{1}{2}p_{xx}$ by direct differentiation.\nNon-example: $p_t = -\\frac{1}{2} p_{xx}$ is not a Fokker–Planck equation, since it would need $D = -\\frac{1}{2}$, which is not positive semidefinite.\nThe equation is also called the forward Kolmogorov equation.\nIn one dimension it reads $p_t = -(\\mu p)_x + (D p)_{xx}$.\n",
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