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  "body": "\n## Definition\n\nLet $(B_t)_{t \\ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] on a [[probability space|0NFPJVJJHX]] $(\\Omega, \\mathcal{F}, P)$.\nThe *local time* of $B$ is a random field $L = (L(t, x))_{t \\ge 0, \\, x \\in \\R}$ such that, almost surely:\n\n1. the map $(t, x) \\mapsto L(t, x)$ is continuous;\n2. for every $t \\ge 0$ and every bounded Borel function $f \\colon \\R \\to \\R$,\n$$\\int_0^t f(B_s) \\, ds = \\int_{\\R} f(x) \\, L(t, x) \\, dx.$$\n\nSuch a field exists, and it is unique up to indistinguishability.\nFor each $t$ and $x$ it satisfies, almost surely,\n$$L(t, x) = \\lim_{\\varepsilon \\downarrow 0} \\frac{1}{2\\varepsilon} \\int_0^t \\mathbf{1}_{\\set{\\abs{B_s - x} \u003c \\varepsilon}} \\, ds.$$\n\n## Remarks\n\nExample: $L(0, x) = 0$ for all $x$, and $E[L(t, 0)] = \\int_0^t (2\\pi s)^{-1/2} \\, ds = \\sqrt{2t/\\pi}$, because the density of $B_s$ at $0$ is $(2\\pi s)^{-1/2}$.\nNon-example: the field $\\int_0^t \\mathbf{1}_{\\set{B_s = x}} \\, ds$ is almost surely $0$ for every $x$, so it fails clause 2 with $f = 1$ and $t \u003e 0$.\nFor fixed $x$, the map $t \\mapsto L(t, x)$ is nondecreasing and increases only on the set of times $t$ with $B_t = x$.\nThe notations $L_t^x$ and $\\ell_t(x)$ are also common.\nWith this normalisation, $L(t, 0)$ has the same law as $\\max_{s \\le t} B_s$.\n",
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