Multiple Points of Sample Paths of Markov Processes

Abstract

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 Levy processes with positive lower indices, solving problems posed by J. P. Kahane and S. J. Taylor.