## $$p$$-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)

### MSC:

 60J55 Local time and additive functionals

### Keywords:

local time; stable process; Brownian local time

### Citations:

Zbl 0745.00038; Zbl 0476.60046; Zbl 0492.60076; Zbl 0465.60065