 Brownian local time

For Brownian motion with generator $\frac{1}{2}\Delta$, 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 $|B_t-a|=|B_0-a|+{\int }_0^t\mathrm{sgn}(B_s-a)d{B}_s+L_t^a$. The symmetric-window approximation is $(2\epsilon {)}^{-1}{\int }_0^t{1}_{\{|B_s-a|<\epsilon \}}ds$. 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.