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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 (1976), unpublished].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

##### MSC:
 60J99 Markov processes 60J45 Probabilistic potential theory
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