engleski

Harnack Inequalities for some Levy Processes

In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in $\R^d$. We give several examples when the scale invariant Harnack inequality does not hold. For any $\alpha\in (0, 2)$ we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric L\' evy process in $\R^d$ with a L\' evy density given by $c|x|^{; ; ; ; -d-\alpha}; ; ; ; 1_{; ; ; ; \{; ; ; ; |x|\leq 1\}; ; ; ; }; ; ; ; +j(|x|)1_{; ; ; ; \{; ; ; ; |x|>1\}; ; ; ; }; ; ; ; $, where $0\leq j(r)\leq cr^{; ; ; ; -d-\alpha}; ; ; ; $, $\forall r>1$, for some constant $c$. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion with subordinator with Laplace exponent $\phi(\lambda)=\lambda^{; ; ; ; \alpha/2}; ; ; ; \ell(\lambda)$, $\lambda>0$, where $\ell$ is a slowly varying function at infinity and $\alpha\in (0, 2).$

Harnack inequality; random walk; Green function; Poisson kernel; stable process; harmonic function; subordinator; subordinate Brownian motion

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