Celebratio Mathematica — Chung

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K. L. Chung and R. Durrett: “Downcrossings and local time,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 : 2 (1976), pp. 147–149. MR 0405605 Zbl 0348.60111 article

Let \( \{W(t):t\geq 0\} \) be the standard Brownian motion with all paths continuous. Let \( M(t)= \max_{0\leq s\leq t}W(s) \) be the maximum process and \( Y(t)=M(t)-W(t) \) be reflecting Brownian motion. If \( d_{\varepsilon}(t) \) is the number of times \( Y \) crosses down from \( \varepsilon \) to 0 before time \( t \), then it was Paul Lévy’s idea that \begin{equation*}\tag{1} P\bigl\{\lim_{\varepsilon\to 0} \varepsilon d_{\varepsilon}(t) = M(t),\ \forall t\geq 0\bigr\} = 1. \end{equation*} In [1974] Itô and McKean demonstrated the almost sure convergence of \( \varepsilon d_{\varepsilon}(t) \) using martingale methods. To identify the limit they used the hard fact, due to Lévy, that \begin{equation*}\tag{2} P\Bigl\{\lim_{\varepsilon\to 0} \frac{ \text{measure} \{s: Y(s) < \varepsilon,\ s\leq t\} }{2\varepsilon}=M(t), \ \forall t\geq 0\Bigr\} = 1 \end{equation*} and computed the second moment of the difference of the expressions in (1) and (2). In this paper, by examining the excursions in Brownian motion and using a new formula for the distribution of their maxima, we obtain a direct identification of the limit in (1) without using (2).

K. L. Chung, R. Durrett, and Z. X. Zhao: “Extension of domains with finite gauge,” Math. Ann. 264 : 1 (1983), pp. 73–79. MR 709862 Zbl 0519.60080 article

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