Harnack inequalities for subordinate Brownian motions

Abstract

In this paper, we consider subordinate Brownian motion  $X$ in $\mathbb{R}^d$, $d \ge 1$,  where the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions. The scale invariant Harnack inequality  is proved for $X$.   We first give new forms of asymptotical properties of the Lévy and potential density of the subordinator near zero. Using these results we find asymptotics of the Lévy density and potential density of $X$ near the origin, which is essential to our approach. The examples which are covered byour results include geometric stable processes and relativistic geometric stable processes, i.e. the cases when the subordinator has the Laplace exponent\[\phi(\lambda)=\log(1+\lambda^{\alpha/2})\  (0<\alpha\leq 2)\]and\[\phi(\lambda)=\log(1+(\lambda+m^{\alpha/2})^{2/\alpha}-m)\ (0<\alpha<2,\,m>0)\,.\]