Brownian motion normalized by maximum local time
Let \(W\) be a standard one-dimensional Brownian motion, \(L(t,x)\) the jointly continuous local time for \(W\), and set \(W^{*}(t) = \sup _{0\leq s\leq t}|W(s)|\), \(t\geq 0\). \textit{F. B. Knight} [Ann. Probab. 1, 1026-1034 (1973; Zbl 0271.60088)] proved that if \(\varphi (t)/t\) is a positive nondecreasing function, then the probability \(P\{W^{*}(t)>\varphi (L(t,0)) \;\text{infinitely often}\}\) is either one or zero; the former possibility occurs if and only if the integral \(\int ^{\infty } 1/\varphi (t) dt\) diverges. An analogous result for \(P\{W^{*}(t)<\psi (L(t,0)) \;\text{infinitely often}\}\) was obtained recently by \textit{D. Khoshnevisan} [Electron. J. Probab. 1, No. 1, 1-18 (1996)]. In the paper under review, the author finds counterparts to Knight's and Khoshnevisan's theorems with the local time \(L(t,0)\) at zero replaced with the maximum local time \(L^{*}(t) = \sup _{x\in \mathbb R} L(t,x)\); the obtained results imply that normalizations via \(L(\cdot ,0)\) and \(L^{*}\) belong to different Lévy classes for the Wiener process \(W\). For example, the following result (Theorem 1.1) is established: With the same \(\varphi \) as above, \(W^{*}(t)>\varphi (L^{*}(t))\) infinitely often with probability one if and only if \(\int ^{\infty } t^{-2}\varphi (t)\exp \bigl (-(2t) ^{-1}j^{2}_{0}\varphi (t)\bigr ) dt=\infty \), where \(j_{0}\) is the smallest positive zero of the Bessel function \(J_{0}\). If the integral converges, then the above probability is zero.
scientific article; zbMATH DE number 1134798
scientific article