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A standard Brownian motion starts at zero, has continuous sample paths, independent stationary increments, and $W_t-W_s\sim \mathcal{N}(0,t-s)$ for $t>s$. The Brownian reflection principle reflects the increments after the first hitting time of a positive level and preserves Brownian law, by symmetry and the strong Markov property.

Reflect after the first hit of $x$. On the event that this hit occurs by time one, $W_1$ changes to $2x-W_1$. This is a measure-preserving involution taking the event $W_1\le x-y$ to $W_1\ge x+y$, proving the specified identity for $x>0,y\ge 0$; the $x=0$ case follows by symmetry. With $y=0$, the maximum event splits, up to zero-probability equality, into reflected equal halves. ThusHence $M$ and $|W_1|$ have the same distribution, and $M>0$ almost surely.

For the ratio we need the joint law, not just these marginals. For $w<m$, the reflected identity gives $\mathbb{P}(M\ge m,W_1\le w)=\mathbb{P}(W_1\ge 2m-w)$. Differentiating yieldswhere $\phi$ is the standard normal density. Put $w=rm$; the change of variables has Jacobian $m$. Therefore, for $r<1$,The density is zero for $r>1$ and has no atom at one. Its integral on $(-\infty ,1)$ is one; negative ratios have no lower bound.

This gives the complete Brownian terminal-to-maximum ratio law.