Distribution of the supremum of a Brownian sheet

Question

We know that for a Brownian motion $B$, $\sup_{0\leq t\leq 1} B(t)$ has same law as $|Z|$, where $Z\sim \mathcal{N}(0,1)$. Now if $B$ is a Brownian sheet, do we know the distribution of $\sup_{0\leq s,t\leq 1} B(s,t)$? Is it true, at least, that the tail of this distribution has an exponential decay?

Answer 1

Let $S:=\sup_{0\le s,t\le 1}B(s,t)$. Then for $\lambda\ge 0$, $$ \mathsf{P}(S\ge \lambda)\le 4\Phi(-\lambda)\le \frac{4\phi(\lambda)}{\lambda}, $$ where $\Phi$ and $\phi$ are the standard normal cdf and pdf, respectively (see Theorem 3 here).