Lower Bounds of the Hausdorff dimension for Feller processes (1406.3849v2)

Abstract: Let $(X_t){t\ge0}$ be a Feller process generated by a pseudo-differential operator whose symbol satisfies $|p(\cdot,\xi)|\infty\le c(1+|\xi|^2)$ and $p(\cdot,0)\equiv0.$ 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{\delta>0: \lim_{|\xi|\to \infty} \frac{\inf_{z\in\R^d} \Re p(z,\xi)}{|\xi|^\delta}=\infty\right}. \end{align*}This, along with the upper bound $ \betaupperstar \Dh E$ with \begin{align*} \betaupperstar &:=\inf\left{\delta>0: \lim_{|\xi|\to \infty}\frac{\sup_{|\eta|\le {|\xi|}}\sup_{z\in\R^d} |p(z,\eta)|}{|\xi|^\delta}=0\right} \end{align*} established in B\"{o}ttcher, Schilling and Wang (2014), extends the dimension estimates for L\'{e}vy processes of Blumenthal and Getoor (1961) and Millar (1971) to Feller processes.