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  "body": "\n## Summary\n\nThe local time $L(t, x)$ of a [[Brownian motion|22735FGJV6]] measures how much time the path spends near the level $x$ up to time $t$.\nIt is the density of the occupation measure of the path with respect to Lebesgue measure.\nLocal time is the correction term in [[Tanaka's formula|GEM0ERJ5ZB]], which extends [[Itô's lemma|2J1RVQB0CM]] to the function $\\abs{x}$.\n\n## Explanations\n\n### L1 human\n\nA Brownian path hits each level it visits infinitely often, but the set of times it sits exactly at a level has length zero.\nSo the plain time spent at $x$ is always zero, and it says nothing.\nLocal time instead counts the time spent in a thin band $(x - \\varepsilon, x + \\varepsilon)$ and divides by the width $2\\varepsilon$.\nAs $\\varepsilon$ shrinks, this ratio has a finite positive limit at levels the path has crossed.\nFor a standard Brownian motion started at $0$, the expected local time at $0$ up to time $t$ is $\\sqrt{2t/\\pi}$.\n",
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