engleski

Harnack inequalities for subordinate Brownian motions

We consider a 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 L\' evy and potential density of the subordinator near zero. Using these results we find asymptotics of the L\' evy 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 \[ \phi(\lambda)=\log(1+\lambda^{; ; ; \alpha/2}; ; ; ) (0<\alpha\leq 2) \] and \[ \phi(\lambda)=\log(1+(\lambda+m^{; ; ; \alpha/2}; ; ; )^{; ; ; 2/\alpha}; ; ; -m) (0<\alpha<2, \, m>0)\, . \]}; ; ;

geometric stable process; Green function; Harnack inequality; Poisson kernel; harmonic function; potential; subordinator; subordinate Brownian motion

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