On the infimum of the local time of a Wiener process (Q1112466)

Let \(\{\) W(t), \(t\geq 0\}\) be a standard Wiener process, and let L(x,t) be its jointly continuous local time. Define \[ T_ r=\inf \{t\geq 0;\quad L(0,t)\geq r\}. \] The upper and lower class behaviour of inf L(y,T\({}_ r)\) is investigated, where the infimum is taken on an interval, which is an appropriately chosen function of r.