Harnack Inequalities for Subordinate Brownian Motions

In this paper, we consider transient subordinate Brownian motion X in R^d, d \geq 1, where the Laplace exponent \phi of the corresponding subordinator satisfies some mild conditions. The scaleinvariant Harnack inequality is proved for X. We first give new forms of asymptotical properties of the Levy and potential density of the subordinator near zero. Using these results we find asymptotics of the Levy density and potential density of X near the origin, which is essential to our approach. The examples which are covered by our 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, d > \alpha) and \phi(\lambda)=\log(1+(\lambda+m^{\alpha/2})^{2/\alpha}-m) (0<\alpha<2,\,m>0,d >2).