Small ball estimates for Brownian motion under a weighted sup-norm (Q2714374)

Let \(\{W(t);0\leq t\leq 1\}\) be a real-valued Wiener process starting form 0 and \(\|. \|\) be a semi-norm in the space of real functions on \([0,1]\). This paper contains the proof of the following result along with some interesting applications: If a positive function, i.e. \(\inf_{\delta \leq t\leq 1}f(t)>0\) for all \(0<\delta \leq 1\), satisfies either \(\inf_{0\leq t\leq 1}f(t)>0\) or \(f\) is non-decreasing in a neighbourhood of 0, then NEWLINE\[NEWLINE \lim_{\varepsilon\to 0} \varepsilon^2 \log \mathbb P(\|W(t)\|_f<\varepsilon)= -\frac{\pi^2} 8 \int^1_0\frac {dt}{f^2(t)}NEWLINE\]NEWLINE where \(\|W\|_f\overset{\text{def}} = \sup_{0\leq t\leq 1} \frac{|W(t)|}{f(t)}\).