Let $B$ a Brownian motion. Show that for all $t_0$,$$p\{t_0\text{ is a local maximum for }B\}=0$$ but a.s. local maximal are a countable dense set in $(0,\infty )$.
For the first part, $$p\{t_0\text{ is a local maximum for }B\}=p\{\exists \delta>0\mid \forall t: |t-t_0|<\delta, B_t\leq B_{t_0}\}$$
So I define $\{S_t=B_t-B_{t_0}\}_{t>0}$ which is also a Brownian. Then, I have to show that $$p\{\exists \delta>0\mid \forall t, |t|<\delta, S_t\leq 0 \}$$ but it doesn't really help. Any idea?
For fixed $t_0 \geq 0$, we have
$$\{\omega; t_0 \, \text{is not a local maximum for $t \mapsto B_t(\omega)$}\} \supseteq \{\omega; \forall \delta>0 \exists t \in [t_0,t_0+\delta]: B_{t}(\omega)>B_{t_0}(\omega)\}.$$
If we set $W_s := B_{s+t_0}-B_{t_0}$, $s \geq 0$, then we can rewrite this as
$$\begin{align*} \{\omega; t_0 \, \text{is not a local maximum for $t \mapsto B_t(\omega)$}\} &\supseteq \{\omega; \forall \delta >0 \exists s \in [0,\delta]: W_s(\omega)>0\} \\ &= \bigcap_{k \in \mathbb{N}} \{\omega; \exists s \in [0,1/k]: W_s(\omega)>0\}. \end{align*}$$
The assertion follows if we can show that
$$\mathbb{P}(\exists s \in [0,1/k]: W_s>0) =1 \tag{1}$$
for all $k \in \mathbb{N}$. It is well-known that
$$M_t := \sup_{s \in [0,t]} W_s \sim |W_t|$$
for any Brownian motion $(W_t)_{t \geq 0}$ (this is e.g. a consequence of the reflection principle). Therefore,
$$\mathbb{P}(\exists s \in [0,1/k]: W_s>0)= \mathbb{P}(M_{1/k}>0) = \mathbb{P}(|B_{1/k}|>0) = 1,$$
and this proves $(1)$.