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  "body": "\n## Summary\n\nThe Kolmogorov backward equation is a parabolic partial differential equation satisfied by expectations $u(t, x) = E[f(X_t) \\mid X_0 = x]$ of a [[diffusion process|9HDQ094ADQ]].\nIts operator is the [[infinitesimal generator|VPA41PWRR1]] of the diffusion, acting on the starting point $x$.\nIt links stochastic differential equations with partial differential equations and is the basis of many probabilistic formulas for solutions of such equations.\n\n## Explanations\n\n### L1 human\n\nStart a random process at the point $x$ and ask for the average value of $f(X_t)$ at a later time $t$.\nThe answer is a function $u(t, x)$ of the time and the starting point.\nThe backward equation says how $u$ changes with $t$: by the drift and the diffusion acting on the starting point.\nFor a Brownian motion, $u$ solves the heat equation $\\partial_t u = \\tfrac12 \\partial_{xx} u$.\nFor instance, $f(y) = y^2$ gives $u(t, x) = x^2 + t$, and both sides equal $1$.\n",
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