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  "body": "\n## Summary\n\nThe infinitesimal generator of a Markov process is the operator that gives the rate of change of expected values at time $0$.\nIt is the derivative at $t = 0$ of the transition [[semigroup|0XMEM406JM]] of the process.\nFor a [[Feller process|4HS3C12472]] the generator determines the law of the process.\nIt links Markov processes with differential operators: Brownian motion has generator $\\tfrac12 \\Delta$.\n\n## Explanations\n\n### L1 human\n\nTake a function $f$ of the state and ask how fast its average value changes when the process has just started at $x$.\nThe answer, as a function of $x$, is $Af(x)$, and $A$ is the generator.\nFor a chain with finitely many states, the generator is the matrix of jump rates.\nFor a particle that jumps from $x$ to $x + 1$ at rate $\\lambda$, one gets $Af(x) = \\lambda (f(x+1) - f(x))$.\nFor Brownian motion it is half the second derivative, which is why the heat equation appears.\n",
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