Lower bounds of the Hausdorff dimension for the images of Feller processes

Abstract

Let (Xt)t⩾0 be a Feller process generated by a pseudo-differential operator whose symbol satisfies ‖p(⋅,ξ)‖∞⩽c(1+|ξ|2) and p(⋅,0)≡0. We prove that, for a large class of examples, the Hausdorff dimension of the set {Xt:t∈E} for any analytic set E⊂[0,∞) is almost surely bounded below by δ∞dimHE, whereδ∞≔sup{δ>0:lim|ξ|→∞infz∈RdRep(z,ξ)|ξ|δ=∞}. This, along with the upper bound β∞dimHE with β∞≔inf{δ>0:lim|ξ|→∞sup|η|⩽|ξ|supz∈Rd|p(z,η)||ξ|δ=0} 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.