Let $X_t$ be a non-standard Brownian motion with $X_t \sim \mathcal N(0,\sigma^2 t)$ where $\sigma$ is the standard deviation of the Wiener process.
I am wondering whether there are any estimates on the process
$$M_T:=\sup_{t \in [0,T]} \vert X_t \vert$$
known? In other words, can we estimate
$$\mathcal P(M_T \ge a)$$ for any $a >0$?