## $$p$$-variation of the local times of symmetric stable processes and of Gaussian processes with stationary increments.(English)Zbl 0762.60069

Let $$X=(X_ t, t\in R_ +)$$ be a symmetric stable process of order $$1<\beta\leq 2$$. Then $$X$$ has jointly continuous local times $$L=(L^ x_ t, (t,x)\in R_ +\times R)$$. This paper is concerned with the $$p$$- variation of the process in the space variable $$L^{\cdot}_ t$$, where $$t>0$$ is a fixed time. The first result is the following. Consider a sequence $$\{\pi(n)\}$$ of partitions of $$[0,a]$$, and denote by $$m(\pi(n))$$, the mesh of $$\pi(n)$$. The authors prove that if $$m(\pi(n))$$ tends to 0, then the $$2/(\beta-1)$$-variation of $$L^{\cdot}_ t$$ along $$\pi(n)$$ converges in $$L^ r$$ to $$c(\beta)\int^ a_ 0| L^ x_ t|^{1/(\beta-1)}dx$$, where $$c(\beta)$$ denotes a constant depending explicitly on $$\beta$$. The convergence also holds almost surely whenever $$m(\pi(n))=o(1/\log n)^{1/(\beta-1)}$$. A closely related result was proved previously by N. Bouleau and M. Yor [C. R. Acad. Sci., Paris, Sér. I 292, 491-494 (1981; Zbl 0476.60046)] in the Brownian case.
The second result concerns the $$\psi$$-variation of the local times, with $\psi(x)=| x/\sqrt{2 \log\log 1/x}|^{2/(\beta-1)}.$ Specifically, the supremum of the $$\psi$$-variation of $$L^{\cdot}_ t$$ along all the partitions of $$[0,a]$$ with mesh larger than $$\delta$$ converges almost surely as $$\delta$$ tends to 0 to $$c'(\beta)\int^ a_ 0| L^ x_ t|^{1/(\beta-1)}dx$$, where $$c'(\beta)$$ denotes another constant depending on $$\beta$$.
The proofs rely in part on a key relation between the sample path of $$L$$ and that of $$G$$, where $$G$$ is the so-called associated Gaussian process, see the paper reviewed above. The main task in this paper consists of proving new results on the $$p$$-variation of stationary Gaussian processes, which are of independent interest.
Reviewer: J.Bertoin (Paris)

### MSC:

 60J55 Local time and additive functionals 60G15 Gaussian processes 60G17 Sample path properties

Zbl 0476.60046
Full Text: