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Multiple points of sample paths of Markov processes. (English) Zbl 0753.60036

Summary: We show that certain \(d\)-dimensional Markov processes \(X(t)\), \(t\geq 0\), have the property that if \(E\) is a closed subset of \(R_ +\) with sufficiently large Hausdorff dimension, then \(X(E)\) has \(k\)-multiple points. This is applied directly to diffusions driven by stochastic differential equations and Lévy processes with positive lower indices, solving problems posed by J.-P. Kahane [Publ. Math. Orsay 83-02, 74-105 (1983; Zbl 0512.60069) and C. R. Acad. Sci., Paris, Sér I 295, 531-534 (1982; Zbl 0503.60081)] and S. J. Taylor [Math. Proc. Camb. Philos. Soc. 100, 383-406 (1986; Zbl 0622.60021)].

MSC:

60G17 Sample path properties
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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