Feller processes generated by pseudo-differential operators: On the Hausdorff dimension of their sample paths
Let \((X_t)_{t\geq 0}\) be a Feller process with state space \(\mathbb{R}^n\). The corresponding generator is under minimal assumptions on its domain a pseudo-differential operator with symbol \(-q(x,\xi)\), \(q:\mathbb{R}^n \times \mathbb{R}^n \to\mathbb{C}\) is a locally bounded function with respect to \(x\) and a continuous negative definite function with respect to \(\xi\). Suppose that for a fixed continuous negative definite function \(\psi\) we have the estimate \(\sup_{x\in \mathbb{R}^n} | q(x,\xi) |\leq c(1+ \psi(\xi))\). The Hausdorff dimension of the set \(\{X_t;t\in E\}\), \(E\subset [0,1]\) is any analytic set, is almost surely bounded above by \(\beta_\psi \dim E\). Here \(\beta_\psi\) is the Blumenthal-Getoor upper index of the Lévy process associated with \(\psi(\xi)\), and \(\dim E\) is the Hausdorff dimension of \(E\).
scientific article; zbMATH DE number 1208786