= Brownian local time
{c}
{title2=$L_t^a$}
= Local time of Brownian motion
{synonym}
For <Brownian motion> with generator $\tfrac12\Delta$, its local time is the occupation density normalized by $\int_0^t h(B_s)ds=\int_{\mathbb R}h(a)L_t^a\,da$ for nonnegative Borel $h$. In the <Tanaka formula> it satisfies $|B_t-a|=|B_0-a|+\int_0^t\operatorname{sgn}(B_s-a)dB_s+L_t^a$. The symmetric-window approximation is $(2\varepsilon)^{-1}\int_0^t\mathbf1_{\{|B_s-a|<\varepsilon\}}ds$. The factor two is essential. <Occupation approximations for Brownian local time> give a direct derivation at level zero from smooth approximations and maximal martingale estimates.
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