A note on the continuity of local times

Abstract:

Several conditions are given for a stochastic process $X(t)$ on $[0,1]$ to have a local time which is continuous in its time parameter (for example, in the Gaussian case, the integrability of ${[E{(X(t) - X(s))^2}]^{ - 1/2}}$ over the unit square). Furthermore, for any Borel function $F$ on $[0,1]$ with a continuous local time, the approximate limit of $|F(s) - F(t)|/|s - t|$ as $s \to t$ is infinite for a.e. $t \in [0,1]$ and $s|F(s) = F(t)$ is uncountable for a.e. $t \in [0,1]$.