On local times of a symmetric stable process as a doubly indexed process

Abstract

We consider the local time process $(L_t^x,x\in ℝ,t\ge 0)$ of a symmetric stable process X with an index β in (1,2]. We compute the p-variation of L on any rectangle of $ℝ\times [0,\mathrm{\infty })$ . Unlike for the p-variation of L with respect to the spatial parameter (studied by Marcus and Rosen), we show here that the Brownian case - when β= 2 - is atypical.