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  "body": "\n## Definition\n\nLet $(E, \\mathcal{E})$ be a measurable space.\nLet $(P_t)_{t \\ge 0}$ be a sub-Markov transition semigroup on $E$.\nThat is, each $P_t$ is a kernel as for a [[Markov kernel|2AVNDP8BJP]], except that the total mass satisfies $P_t(x, E) \\le 1$, $P_0$ is the identity, and $P_s P_t = P_{s+t}$.\nFor a measurable $f \\colon E \\to [0, \\infty]$, write $P_t f(x) = \\int_E f(y) \\, P_t(x, dy)$.\nA measurable function $f \\colon E \\to [0, \\infty]$ is *excessive* for $(P_t)$ if:\n\n1. $P_t f \\le f$ for every $t \\ge 0$.\n2. $P_t f(x) \\to f(x)$ as $t \\downarrow 0$, for every $x \\in E$.\n\n## Remarks\n\nUnder clause 1, $t \\mapsto P_t f$ is decreasing, since $P_{t+s} f = P_t (P_s f) \\le P_t f$; so the limit in clause 2 exists and is at most $f$.\nA function with clause 1 alone is called supermedian; for $\\alpha \u003e 0$, the $\\alpha$-excessive functions are those that are excessive for the semigroup $e^{-\\alpha t} P_t$.\nExample: let $(P_t)$ be the semigroup of [[Brownian motion|22735FGJV6]] in $\\R^3$ and $f(x) = 1/\\abs{x}$, with $f(0) = \\infty$.\nThen $f$ is a nonnegative [[superharmonic function|X4V2S56YQQ]], and averaging it against the Gaussian kernel gives $P_t f \\le f$ with $P_t f \\uparrow f$ as $t \\downarrow 0$, so $f$ is excessive.\nNon-example: for the same semigroup, let $f$ be the indicator of the single point $0$.\nThen $P_t f = 0 \\le f$, so clause 1 holds, but $P_t f(0) = 0$ does not tend to $f(0) = 1$, so clause 2 fails.\nFor Brownian motion in $\\R^d$, the excessive functions are exactly the nonnegative superharmonic functions on $\\R^d$, together with the constant $+\\infty$.\n",
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