\(p\)-variation of the local times of symmetric stable processes and of Gaussian processes with stationary increments.

*(English)*Zbl 0762.60069Let \(X=(X_ t, t\in R_ +)\) 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.

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)