17-JUL-2024
We construct Brownian local time $L_t^x$ as the density of the occupation measure of standard Brownian motion and establish three foundational results: the occupation time formula, Tanaka’s formula extending Itô’s lemma to $|B_t – a|$, and Lévy’s representation theorem identifying $L_t^0$ in distribution with $|B_t|$. The entire development is motivated by a single problem in quantum mechanics: the Schrödinger operator $H = -\tfrac{1}{2}\partial_{xx} + \alpha\delta$ requires, via the Feynman–Kac formula, a rigorous interpretation of $\int_0^\tau \delta(B_s)\,ds$ — which is precisely the local time $L_\tau^0$. The quantum consequences follow as direct corollaries: the Feynman–Kac weight for the delta potential is $e^{-\alpha L_\tau^0}$, and the bound state energy $E_0 = -\alpha^2/2$ (for $\alpha < 0$) is derived from the Laplace transform of $L_t^0$ established via Lévy's theorem.