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  "body": "\n## Definition\n\nLet $E$ be a locally compact Hausdorff space with a countable base.\nLet $C_0(E)$ be the [[Banach space|06GZ956657]] of continuous functions on $E$ that vanish at infinity, with the sup norm.\nLet $(X_t)_{t \\ge 0}$ be a [[Feller process|4HS3C12472]] on $E$ with semigroup $T_t f(x) = \\mathbb{E}_x[f(X_t)]$.\nThe *infinitesimal generator* of $X$ is the linear operator $A$ with domain\n$$D(A) = \\set{f \\in C_0(E) : \\lim_{t \\downarrow 0} \\frac{T_t f - f}{t} \\text{ exists in } C_0(E)},$$\ngiven by\n$$Af = \\lim_{t \\downarrow 0} \\frac{T_t f - f}{t} \\quad \\text{for } f \\in D(A),$$\nwhere the limit is in the sup norm.\n\n## Remarks\n\nExample: for a continuous-time Markov chain on a finite set $S$ with rate matrix $Q$, one has $T_t = e^{tQ}$, so $(T_t f - f)/t \\to Qf$ for every $f$, and $A = Q$ with $D(A)$ the whole space.\nExample: for Brownian motion on $\\R^d$, every twice continuously differentiable $f$ with $f$, its first and its second derivatives in $C_0(\\R^d)$ lies in $D(A)$, and $Af = \\tfrac12 \\Delta f$.\nNon-example: for Brownian motion on $\\R$, the function $f(x) = e^{-x^2} \\abs{x}$ is in $C_0(\\R)$ but not in $D(A)$, since it is not differentiable at $0$.\nFor $f \\in D(A)$, Dynkin's formula says that $f(X_t) - f(X_0) - \\int_0^t Af(X_s)\\, ds$ is a martingale.\nThe semigroup satisfies $\\frac{d}{dt} T_t f = A T_t f = T_t A f$ for $f \\in D(A)$, the [[backward|KQMJ7M57ER]] and forward equations.\n",
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