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Small ball estimates for Brownian motion under a weighted sup-norm. (English) Zbl 0973.60083
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 $\lim_{\varepsilon\to 0} \varepsilon^2 \log \mathbb P(\|W(t)\|_f<\varepsilon)= -\frac{\pi^2} 8 \int^1_0\frac {dt}{f^2(t)}$ where $$\|W\|_f\overset{\text{def}} = \sup_{0\leq t\leq 1} \frac{|W(t)|}{f(t)}$$.

##### MSC:
 60J65 Brownian motion
##### Keywords:
small ball problem; Brownian motion
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