Potential Theory of Geometric Stable Processes
Potential Theory of Geometric Stable Processes
复制标题
几何稳定过程势理论
DOI：
10.1007/s00440-005-0470-3
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发表时间：
2006
影响因子：
2
通讯作者：
Z. Vondraček
中科院分区：
文献类型：
--
作者：
H. Šikić;R. Song;Z. Vondraček
本文通过将对称几何稳定过程实现为具有几何稳定从属的布朗运动，研究了对称几何稳定过程的势理论。更准确地说，我们建立了对称几何稳定过程的格林函数和lsamvy密度的渐近性质。这些函数在零附近的渐近性表现出与稳定过程的渐近性非常不同的特征。Green函数在接近零的情况下表现为1/(|x|d log 2|x|)，而lsamvy密度表现为1/|x|d。我们还研究了具有迭代几何稳定从属布朗运动的格林函数和lsamvy密度的渐近行为。作为一个应用，我们建立了对这些过程的小球容量的估计，以及小球的平均退出时间估计和这些过程的Harnack不等式。
In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(|x|d log 2|x|), while the Lévy density behaves like 1/|x|d. We also study the asymptotic behaviors of the Green function and Lévy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes.