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

## TL;DR

This paper develops a rough path framework for integrating with respect to the local time of symmetric alpha-stable processes, extending beyond Young's integral to handle less smooth functions.

## Contribution

It establishes a rough path theory for local times of symmetric alpha-stable processes, enabling integration for functions with lower regularity.

## Key findings

- Proves local time has bounded p-variation for p > 2/(-1)
- Defines local time integral as a Young integral for certain q-variations
- Extends integration theory using rough paths for q in [2/(3-), 4)

## Abstract

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 $\int_{-\infty}^{\infty}\triangledown_-^{\alpha-1}f(x)d_x L_t^x$ as a Young integral for less smooth functions being of bounded $q$-varition with $1\leq q<\frac{2}{3-\alpha}$. When $q\geq \frac{2}{3-\alpha}$, Young's integration theory is no longer applicable.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 $\frac{2}{3-\alpha}\leq q< 4$.