Let be the jointly continuous local times of a Brownian motion W. Show Show the null set can be taken to be independent of y. Thus there is no need to take a subsequence εn to get almost sure...

Let

be the jointly continuous local times of a Brownian motion W. Show

Show the null set can be taken to be independent of y. Thus there is no need to take a subsequence ε_n
to get almost sure convergence to

Let W be a Brownian motion and fix t. Show that the function

is continuous, a.s., but that the function

is not continuous.

Chapter 15