On the distribution of supremum of diffusion local time
\((X(t), P_ x)_{t\geq 0}\) is a one-dimensional regular diffusion on an interval with transitions \(p(t,x,y)\) and local time \(L_{t,x}\). The law of \(\sup_{t\geq 0}(L_{t,x}-\mu t)\), \(\mu\geq 0\), is computed to be under \(P_ x\) an exponential distribution with parameter \(\alpha^*(x)/\mu\), where \(\alpha^*\) is the unique positive solution of \(\alpha/\mu-1/G_ \alpha=0\), where \(G_ \alpha\) is the Green kernel \(\int\exp(-\alpha t)p(t,x,y)=dt\).
scientific article; zbMATH DE number 503673