\(p\)-variation of the local times of stable processes and intersection local time. (English) Zbl 0796.60078

Seminar on stochastic processes, Proc. Semin., Los Angeles/CA (USA) 1991, Prog. Probab. 29, 157-167 (1992).
[For the entire collection see Zbl 0745.00038.]
Let \(L^ x_ t\) be the local time of the symmetric stable process of order \(\beta>1\). The author studies the \(p\)-variation of \(L^ x_ t\) in \(x\) and generalizes results concerning Brownian local time of N. Bouleau and M. Yor [C. R. Acad. Sci., Paris, Sér. I 292, 491-494 (1981; Zbl 0476.60046)] and E. Perkins [Z. Wahrscheinlichkeitstheorie Verw. Geb. 60, 437-451 (1982; Zbl 0465.60065)]. For arbitrary \(\beta>1\) the methods conduct to the following
Theorem (1.2). Let \(\beta>1\), then \[ \sum_{x_ i\in\Pi}\Bigl({L^{x_ i}_ t-L_ t^{x_{i-1}}\over(x_ i-x_{i- 1})^ \gamma}\Bigr )^{2k}\to\bar c\int^ b_ a(L^ x_ t)^ kdx \] in \(L^ 2\), uniformly in both \(t\in[0,T]\) and \(\Pi\in Q(a,b)\) as \(m(\Pi)\to 0\), where \(\gamma=(\beta-1)/2-1/2k\) and \(\bar c=(2k)!!(4c)^ k\), \(c=\int^ \infty_ 0(p_ t(0)-p_ t(1))dt\), and \(p_ t(x)\) is the transient density for the stable process.
Here \(Q(a,b)\), for \(a,b<\infty\), denotes the set of partitions \(\Pi=\{x_ 0=a<x_ 1\cdots<x_ n=b\}\) of \([a,b]\), and \(m(\Pi)=\sup_ i(x_ i-x_{i-1})\) is used to denote the mesh size of \(\Pi\).
Reviewer: G.Orman (Braşov)


60J55 Local time and additive functionals