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  "body": "\n## Definition\n\nLet $b \\colon \\R^n \\to \\R^n$ and $\\sigma \\colon \\R^n \\to \\R^{n \\times m}$ be functions, and let $a = \\sigma \\sigma^{T}$.\nConsider the [[diffusion|9HDQ094ADQ]] given by the stochastic differential equation $dX_t = b(X_t) \\, dt + \\sigma(X_t) \\, dB_t$, where $B$ is an $m$-dimensional [[Brownian motion|22735FGJV6]].\nIts [[generator|VPA41PWRR1]] is the differential operator\n$$\\mathcal{L} g(x) = \\sum_{i=1}^n b_i(x) \\, \\partial_i g(x) + \\frac12 \\sum_{i,j=1}^n a_{ij}(x) \\, \\partial_i \\partial_j g(x),$$\nwith [[partial derivatives|09EGC6F8PT]] $\\partial_i$ in the variable $x$.\nThe *Kolmogorov backward equation* of the diffusion is the equation\n$$\\partial_t u(t, x) = \\mathcal{L} u(t, x), \\quad t \u003e 0, \\ x \\in \\R^n,$$\nfor a function $u \\colon [0, \\infty) \\times \\R^n \\to \\R$, with an initial condition $u(0, x) = f(x)$.\n\n## Remarks\n\nUnder regularity conditions on $b$, $\\sigma$ and $f$, the function $u(t, x) = E[f(X_t) \\mid X_0 = x]$ solves the backward equation.\nIf $X$ has transition densities $p(t, x, y)$, they satisfy $\\partial_t p = \\mathcal{L}_x p$, where $\\mathcal{L}_x$ acts on the starting point $x$.\nThe forward, or Fokker–Planck, equation acts instead on the end point $y$.\nExample: for the [[Ornstein–Uhlenbeck process|HZM2ASCDY4]] $dX = -X \\, dt + dB$ in $\\R$ and $f(y) = y$, one has $u(t, x) = x e^{-t}$.\nCheck: $\\partial_t u = -x e^{-t}$ and $\\mathcal{L} u = -x \\, \\partial_x u + \\tfrac12 \\partial_{xx} u = -x e^{-t}$.\nNon-example: for the same process, $v(t, x) = x e^{t}$ has $\\partial_t v = x e^{t}$ but $\\mathcal{L} v = -x e^{t}$, so it does not solve the equation.\nThe equation follows from [[Itô's lemma|2J1RVQB0CM]] applied to $s \\mapsto u(t - s, X_s)$.\n",
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