zbMATH — the first resource for mathematics

Multiple points of Lévy processes. (English) Zbl 0684.60057
Let \(X_ t\) be a Lévy process in \(R^ d\) with transition density \(p_ t(x)\) and characteristic function \(E(e^{iy\cdot X_ t})=e^{- t\psi (y)}\). The authors prove the existence of k-multiple points of X under the following hypotheses (A) and (B): \[ (A)\quad \int_{| x| \leq \epsilon}(\int^{T}_{0}p_ t(x)dt)^ kdx<\infty \quad for\quad some\quad \epsilon,\quad T>0. \] \[ (B)\quad \int^{T}_{0}p_ t(0)dt>0. \] This result proves a conjecture of W. J. Hendricks and S. J. Taylor [Concerning some problems about polar sets of processes with independent increments. (1976), unpublished]. By using this result, the authors also investigate the Hausdorff dimension of the set of k-multiple points \(E_ k=\{(t_ 1,t_ 2,...,t_ k)\); \(X_{t_ 1}=X_{t_ 2}=...=X_{t_ k}\}\). The result says that \[ \dim E_ k\geq k-(k-1)d/\beta '',\quad where\quad \beta ''=\sup \{\alpha \geq 0;\quad | y|^{-\alpha}Re \psi (y)\to 0\quad as\quad | y| \to 0\}. \]
Reviewer: Y.Oshima

60J99 Markov processes
60J45 Probabilistic potential theory
Full Text: DOI