engleski

Harnack Inequalities for some Lévy Processes

Harnack inequality for nonnegative functions which are harmonic with respect to random walks in $\R^d$ is proved. Several examples when the scale invariant Harnack inequality does not hold are given. For any $\alpha\in (0, 2)$ we also prove the scale invariant Harnack inequality for non-negative 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 \[0\leq j(r)\leq cr^{; ; -d-\alpha}; ; \textrm{; ; for all }; ; r>1.\] We establish the scale invariant Harnack inequality for non-negative 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 $\alpha\in (0, 2).$ Finally, we consider a subordinate Brownian motion where the corresponding subordinator has the Laplace exponent \[ \psi(\lambda)=\frac{; ; \lambda}; ; {; ; \log(1+\lambda)}; ; -1, \ \lambda>0. \] The scale invariant Harnack inequality and logarithmic Sobolev inequality are proved for this process. We obtain an upper bound of heat kernel $p(t, x, y)$ for small $t>0$ and $|x-y|$.

Harnack Inequalities for some Lévy Processes

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