Paul Lévy’s Way to His Local Time

Abstract

In his 1939 paper [1] Lévy introduced the notion of local time for Brownian motion. He gave several equivalent definitions, and towards the end of that long paper he proved the following result. Let ∈ > 0, t > 0, B(0) = 0, $${L_ \in }\left( t \right) = m\left\{ {s \in \left[ {0,t} \right]\left| {0 < B\left( s \right) < \in } \right.} \right\}/ \in $$ where B(t) is the Brownian motion in R and m is the Lebesgue measure. Then almost surely the limit below exists for all t >0: $$\begin{array}{*{20}{c}} {\lim } \\ { \in \to 0} \end{array}{L_ \in }\left( t \right) = L\left( t \right).$$ (0.2)