Harnack Inequalities for some Lévy Processes
Harnack Inequalities for some Lévy Processes
复制标题
某些 Lévy 过程的 Harnack 不等式
DOI：
10.1007/s11118-009-9153-5
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发表时间：
2010
影响因子：
1.1
通讯作者：
Ante Mimica
中科院分区：
文献类型：
--
作者：
Ante Mimica
本文证明了关于随机游动的调和非负函数的Harnack不等式。我们给出了几个例子时，规模不变Harnack不等式不成立。对于任意α ∈（0，2），我们还证明了非负调和函数关于Lévy密度为$c的对称Lévy过程的Harnack不等式|X| ^{-d-\alpha}1| X|\leq 1\}}+j（|X|）1_{|X|>1\}}$，其中0 ≤ j（r）≤ cr − d − α，对于某个常数c，<$r > 1。最后，建立了关于服从布朗运动的非负调和函数的Harnack不等式，其中服从布朗运动的拉普拉斯指数为λ（λ）= λα/2 λ（λ），λ > 0，λ是无穷远处的慢变函数，α ∈（0，2）.
In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝd. We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process in ℝd with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$, where 0 ≤ j(r) ≤ cr − d − α, ∀ 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 ϕ(λ) = λα/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2).