Perturbation of symmetric Markov processes
Perturbation of symmetric Markov processes
复制标题
DOI：
10.1007/s00440-007-0065-2
复制
发表时间：
2007-03
影响因子：
2
通讯作者：
Zhen-Qing Chen;P. Fitzsimmons;K. Kuwae;Tusheng Zhang
中科院分区：
文献类型：
--
作者：
Zhen-Qing Chen;P. Fitzsimmons;K. Kuwae;Tusheng Zhang
本文给出了一般对称马氏过程的L2-无穷小生成元的低阶扰动所得到的二次型所对应的半群的路径空间积分表示。一个很有启发性的具体例子是，其中D是有界欧氏区域，是D中的拉普拉斯算子，且Dirichlet边界条件为零，是D中的分数拉普拉斯算子，且外部条件为零。对应的强Markov过程是一个Lévy过程，它是中的布朗运动和一个独立的对称（2s）-稳定过程的和。这种概率表示是Feynman-Kac和Girsanov公式的组合。至关重要的发展是使用的扩展中尾的随机积分为零能量添加剂泛函和相关的伊藤公式，这两个最近开发的陈等人。[Stochastic calculus for Dirichlet processes（preprint）（2006）].
We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of theL2-infinitesimal generatorof a general symmetric Markov process. An illuminating concrete example foris, whereDis a bounded Euclidean domain inis the Laplace operator inDwith zero Dirichlet boundary condition andis the fractional Laplacian inDwith zero exterior condition. The strong Markov process corresponding tois a Lévy process that is the sum of Brownian motion inand an independent symmetric (2s)-stable process inkilled upon exiting the domainD. This probabilistic representation is a combination of Feynman-Kac and Girsanov formulas. Crucial to the development is the use of an extension of Nakao’s stochastic integral for zero-energy additive functionals and the associated Itô formula, both of which were recently developed in Chen et al. [Stochastic calculus for Dirichlet processes (preprint)(2006)].