Lower Bounds of the Hausdorff dimension for Feller processes

Abstract

Let $(X_t{)}_{t\ge 0}$ be a Feller process generated by a pseudo-differential operator whose symbol satisfies $\Vert p(\cdot ,\xi ){\Vert }_{\infty }\le c(1+|\xi {|}^2)$ and $p(\cdot ,0)\equiv 0.$ We prove that, for a large class of examples, the Hausdorff dimension of the set $\{X_t:t\in E\}$ for any analytic set $E\subset [0,\infty )$ is almost surely bounded below by $\betalower \Dh E$, where \begin{align*} \betalower&:=\sup\left\{δ>0: \lim_{|ξ|\to \infty} \frac{\inf_{z\in\R^d} \Re p(z,ξ)}{|ξ|^δ}=\infty\right\}. \end{align*}This, along with the upper bound $ \betaupperstar \Dh E$ with \begin{align*} \betaupperstar &:=\inf\left\{δ>0: \lim_{|ξ|\to \infty}\frac{\sup_{|η|\le {|ξ|}}\sup_{z\in\R^d} |p(z,η)|}{|ξ|^δ}=0\right\} \end{align*} established in Böttcher, Schilling and Wang (2014), extends the dimension estimates for Lévy processes of Blumenthal and Getoor (1961) and Millar (1971) to Feller processes.