A note on the stability of the local time of a Wiener process
Let (L(a,t),a\(\in {\mathbb{R}},t\geq 0)\) be the local time process of a one- dimensional Brownian motion. It is well-known that, for any fixed a, the ratio L(a,t)/\(L(0,t)\) converges to 1 almost surely as t tends to infinity. The purpose of the present paper is to prove the stronger result that, for suitable functions g(t), \[ \lim_{t\to \infty}\sup_{| a| \leq g(t)}| L(a,t)/L(0,t)-1| =0,\quad a.s. \] This result holds for \(g(t)=t^{1/2}(\log t)^{-1}(\log \log t)^{-r}\) when \(r>2\), but not when \(r=1\). Similar statements are obtained when g(t) is replaced by a function of the maximum process of the Brownian motion. The latter results lead to a negative answer to a conjecture of \textit{M. Csörgö} and \textit{P. Révész} [Acta Sci. Math. 48, 85-96 (1985; Zbl 0586.60060)]. The proofs depend on some estimates of \textit{R. F. Bass} and \textit{P. S. Griffin} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 417-436 (1985; Zbl 0554.60076)] and on the celebrated Ray theorem concerning the law of (L(a,\(\tau)\),a\(\in {\mathbb{R}})\), when \(\tau\) is an independent exponential time.
scientific article; zbMATH DE number 4026488
scientific article