= Gaussian maximal bound for Brownian motion
{c}
{title2=$\mathbb P(\sup_{s\le t}|B_s|>\delta)\le2e^{-\delta^2/(2t)}$}
Stop the <Exponential martingale for Brownian motion> $e^{\theta B_s-\theta^2s/2}$ when $B$ first reaches $\delta$ or at $t$. On the hitting event its value is at least $e^{\theta\delta-\theta^2t/2}$, so that event has <probability> at most $e^{-\theta\delta+\theta^2t/2}$. Minimize over $\theta>0$ and apply the same bound to $-B$. The union bound gives the displayed inequality for $t>0$; at zero the <probability> is zero.
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