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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 $f \\colon E \\to \\R$ that vanish at infinity, with the sup norm.\nA *Feller semigroup* on $C_0(E)$ is a family $(T_t)_{t \\ge 0}$ of linear maps $T_t \\colon C_0(E) \\to C_0(E)$ such that:\n\n1. if $f \\in C_0(E)$ and $0 \\le f \\le 1$, then $0 \\le T_t f \\le 1$;\n2. $T_0 = \\id$ and $T_{s+t} = T_s T_t$ for all $s, t \\ge 0$;\n3. $\\norm{T_t f - f} \\to 0$ as $t \\downarrow 0$, for every $f \\in C_0(E)$.\n\nLet $(X_t)_{t \\ge 0}$ be a time-homogeneous [[Markov process|0B77KZCQ51]] in $E$, and let $\\mathbb{E}_x$ be the expectation for the process started at $x$.\nThe process is a *Feller process* if the formula\n$$T_t f(x) = \\mathbb{E}_x[f(X_t)]$$\ndefines a Feller semigroup.\n\n## Remarks\n\nExample: Brownian motion on $\\R$ is a Feller process.\nHere $T_t f(x) = \\int f(y) \\frac{1}{\\sqrt{2\\pi t}} e^{-(y - x)^2/(2t)}\\, dy$; this is in $C_0(\\R)$ when $f$ is, it preserves $0 \\le f \\le 1$, and it tends to $f$ uniformly because $f$ is uniformly continuous.\nNon-example: on $E = \\R$, let the process started at $x$ jump at once to $0$ if $x \u003e 0$ and to $1$ if $x \\le 0$, then stay there.\nThen $T_t f = f(0)$ on $(0, \\infty)$ and $T_t f = f(1)$ on $(-\\infty, 0]$, which is not continuous in $x$, so $T_t$ does not map $C_0(E)$ into itself.\nClause 1 allows $T_t 1 \u003c 1$, so the process may be killed; one then adds a cemetery point to $E$.\nEvery Feller process has a modification with right-continuous paths with left limits, and that modification is strong Markov.\n",
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