Rough path properties for local time of symmetric $\alpha$ stable process

In this paper, we first prove that the local time associated with symmetric $\alpha$-stable processes is of bounded $p$-variation for any $p>\frac{2}{\alpha -1}$ partly based on Barlow’s estimation of the modulus of the local time of such processes.,,The fact that the local time is of bounded $p$-variation for any $p>\frac{2}{\alpha -1}$ enables us to define the integral of the local time $\int_{-\infty}^{\infty}\triangledown_-^{\alpha-1}f(x)d_x L_t^x$asaYoungintegralforlesssmoothfunctionsbeingofbounded$q$-varitionwith$1\leq q<\frac{2}{3-\alpha}$\mathrm{.}When$q\geq \frac{2}{3-\alpha}$, Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric $\alpha$-stable processes for $\frac{2}{3-\alpha }\le q<4$.