$p$-Variation of the Local Times of Symmetric Stable Processes and of Gaussian Processes with Stationary Increments

Abstract

Let $\{L^x_t, (t, x) \in R^+ \times R\}$ be the local time of a real-valued symmetric stable process of order $1 < \beta \leq 2$ and let $\{\pi(n)\}$ be a sequence of partitions of $\lbrack 0, a\rbrack$. Results are obtained for $\lim_{n\rightarrow\infty} \sum_{x_i\in\pi(n)} |L^{x_i}_t - L^{x_{i-1}}_t|^{2/(\beta-1)}$ both almost surely and in $L^r$ for all $r > 0$. Results are also obtained for a similar expression but where the supremum of the sum is taken over all partitions of $\lbrack 0, a\rbrack$ and a function other than a power is applied to the increments of the local times. The proofs use a lemma of the authors' which is a consequence of an isomorphism theorem of Dynkin and which relates sample path behavior of local times with those of associated Gaussian processes. The major effort in this paper consists of obtaining results on the $p$-variation of the associated Gaussian processes. These results are of independent interest since the associated processes include fractional Brownian motion.