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p-variation for families of local times on lines. (English) Zbl 0957.60086

Azéma, J. (ed.) et al., Séminaire de Probabilités XXXIV. Berlin: Springer. Lect. Notes Math. 1729, 171-184 (2000).
The article is devoted to the properties of the local times of the stable processes on the plane. Let \(X_s\) be a two-dimensional symmetric stable process. Let \(e(\theta)=(\cos (\theta), \sin(\theta))\) be the unite vector corresponding to \(\theta .\) Denote by \(L_t ^{a,\theta }\) the local time of the one-dimensional stable process \(X_s\times e(\theta)\) at the point \(a\). Consider the smooth curve \(\gamma :[0,1]\rightarrow R_+ ^1\times [0,2\pi)\). The authors find the limit \(\lim _{n\rightarrow \infty } \sum _{s_i\in \pi (n)} (L_t ^{\gamma (s_i)} -L_t ^{\gamma (s_{i-1})})^{2k}\) for the process of index \(\beta =1+{1\over k}.\)
For the entire collection see [Zbl 0940.00007].

MSC:

60J55 Local time and additive functionals
60H05 Stochastic integrals
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