Lower Bounds of the Hausdorff dimension for Feller processes

TL;DR

This paper establishes lower bounds for the Hausdorff dimension of sets generated by Feller processes, extending classical results for Lévy processes to a broader class of stochastic processes.

Contribution

It introduces new lower bounds for Hausdorff dimensions of Feller process paths, generalizing previous bounds known for Lévy processes.

Findings

Lower bounds depend on the symbol's real part growth rate.

Extends classical Lévy process dimension estimates to Feller processes.

Provides conditions under which the bounds are almost surely attained.

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.