I'm currently working on the maximum of Brownian Motion and have found that the mirror principle can be really helpful with this. I have found and understand a proof of the following theorem:
Given a Brownian Motion $(W(t))_{t\geq0}$, and let $\tau$ be a stopping time with respect to $(\mathscr{F}_t)_{t\geq0}$. Then $$ \begin{equation} W_\tau(t) := \begin{cases} W(t) & 0\leq t\leq \tau \\ 2W(\tau) - W(t) & t > \tau \end{cases} \end{equation} $$ is a Brownian Motion.
In the paper I found for this proof they mentioned that filling in $\tau_x = \text{inf}\{ t \ | \ W(t) = x\} $ gives that $\mathbb{P}(\text{max}\big(W(t)\big) > x) = 2\,\mathbb{P}\big(W(t) > x \big)$. Now, I do understand this intuitively but I cannot prove it. can any of you help me with this? Thanks in advance!!