Reflected Brownian motion with skew symmetric data in a polyhedral domain
Reflected Brownian motion with skew symmetric data in a polyhedral domain
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多面体域中具有斜对称数据的反射布朗运动
DOI：
10.1007/bf00320328
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发表时间：
1987
影响因子：
2
通讯作者：
Ruth J. Williams
中科院分区：
文献类型：
--
作者：
Ruth J. Williams
本文研究了多面体区域上某些反射布朗运动的特征和不变测度。这里所研究的RBM的行为类似于简单多面体内部具有常数漂移μ的d维布朗运动，并且在边界处以取决于被撞击面的方向瞬时反射。在假设反射方向满足Harrison-Williams [9]中首次引入的某个斜对称条件下，证明了这样的RBM可以用一族下鞅来刻画，并且它以概率零到达边界的非光滑部分。在[9]中，解决了与这种RBM相关的纯分析问题。这里，在[9]中得到的指数形式解被证明是RBM的不变测度的密度。此外，如果密度在多面体状态空间上是可积的，则它产生RBM的唯一稳态分布。在这些结果的证明中，RBM的对偶过程和[9]中关于光滑逼近域上反射布朗运动的结果起了关键作用。
SummaryThis paper is concerned with the characterization and invariant measures of certain reflected Brownian motions (RBM's) in polyhedral domains. The kind of RBM studied here behaves like d-dimensional Brownian motion with constant drift μ in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit. Under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced in Harrison-Williams [9], it is shown that such an RBM can be characterized in terms of a family of submartingales and that it reaches non-smooth parts of the boundary with probability zero. In [9], a purely analytic problem associated with such an RBM was solved. Here the exponential form solution obtained in [9] is shown to be the density of an invariant measure for the RBM. Furthermore, if the density is integrable over the polyhedral state space, then it yields the unique stationary distribution for the RBM. In the proofs of these results, a key role is played by a dual process for the RBM and by results in [9] for reflected Brownian motions on smooth approximating domains.