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.