The supremum of Brownian local times on Hölder curves (Q5954101)

One denotes by \(S_{\alpha}\) the closed unit ball of the space of \(\alpha\)-Hölder functions. Moreover we denote by \(L_{t}^{f}\) the local time of space-time Brownian motion on the curve \(f:[0,1]\rightarrow\mathbb R\). The authors prove that the supremum of \(L_{1}^{f}\) over \(f\in S_{\alpha}\) is finite if \(\alpha>1/2\) and infinite if \(\alpha<1/2\).