= Integral lower envelope for the Brownian maximum
{title2=$\liminf_{t\downarrow0}S_t/(\sqrt t\,f(t))=\infty$}
Let $f$ be positive, continuous and nondecreasing, with $\int_0^1f(t)\,dt/t<\infty$. The small-value <probability> for the <Brownian running maximum> is at most a constant times the value. <Brownian scaling> and dyadic integral comparison make the exceptional <probabilities> $\mathbb P(S_{2^{-n-1}}<2^{-n/2}f(2^{-n}))$ summable. The <Borel-Cantelli first lemma> and monotonicity interpolate between dyadic times. Apply this argument to $mf$ for every positive integer $m$ on a countable intersection of probability-one events to obtain the infinite lower limit.
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