= Quadratic variation obstruction to Hölder regularity
{title2=$\sum_{k=0}^{N-1}(w((k+1)/N)-w(k/N))^2\leq C^2N^{1-2\alpha}$}
A function with finite <Hölder seminorm> of exponent $\alpha>1/2$ has squared-increment sum on a uniform $N$-interval partition bounded by $C^2N^{1-2\alpha}$, hence tending to zero. This conflicts with a nonzero quadratic-variation limit. In particular <dyadic quadratic variation of Brownian motion> prevents a Brownian coordinate law supported on such a <Hölder space>. This argument alone gives no conclusion at the critical exponent $1/2$.
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