Basic properties of Brownian local time

Let $W$ be a Brownian motion and $\alpha >0$. Show that

is infinite almost surely if $\alpha \geq 1$ but finite almost surely if $\alpha <1$.

Let $L_{t}^{y}$ 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 $\varepsilon_{n}$ to get almost sure convergence to $L_{t}^{y}$.

Let $W$ be a Brownian motion and fix $t$. Show that the function $x \rightarrow \int_{0}^{t} \; \mathbb {1}_{(-\infty , x]}\left(W_{s}\right) d s$ is continuous, a.s., but that the function $x \rightarrow \; \mathbb {1}_{(-\infty , x]}\left(W_{t}\right)$ is not continuous.

If $L_{t}^{0}$ is the local time of a Brownian motion at 0 , show that $L_{a t}^{0}$ has the same law as $\sqrt{a} L_{t}^{0}$.

Let $W$ be a Brownian motion with local times $L_{t}^{y}$. Set $L_{t}^{*}=\sup_{y} L_{t}^{y}$. Let $p>0$. Prove that there exist constants $c_{1}, c_{2}$ such that if $T$ is any finite stopping time,

The constants $c_{1}, c_{2}$ can depend on $p$, but not on $T$.

Hint: Use Exercise 12.11.