分段光滑边界区域上半鞅反射布朗运动的不变原理
An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries
摘要：
半鞅反射布朗运动（Semimartingale reflecting Brownian motions，SRBM）是一类具有分段光滑边界的域的闭包，由于其作为随机网络的高负荷服务近似，在应用概率中受到广泛关注.本文通过对反射域和反射方向的假设，证明了SRBM的一个扰动结果，即不变原理。这为满足SRBM定义的过程提供了充分条件，除了定义条件中的小随机扰动之外，该过程在分布上接近SRBM。证明这一结果的一个关键因素是一个扰动Skorokhod问题的解的振荡不等式。我们利用不变原理证明了在较弱的条件下SRBM的弱存在性。我们还使用不变原理，结合已知的唯一性结果SRBM，给出了一些充分条件验证近似涉及（i）SRBM在凸多面体与一个常数的反射矢量场的多面体的每一个面，（ii）SRBM在有界区域与分片光滑的边界和可能的非常数反射向量场的边界表面。
英文摘要：
Semimartingale reflecting Brownian motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic approximations for some stochastic networks. In this paper, assuming certain conditions on the domains and directions of reflection, a perturbation result, or invariance principle, for SRBMs is proved. This provides sufficient conditions for a process that satisfies the definition of an SRBM, except for small random perturbations in the defining conditions, to be close in distribution to an SRBM. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. We use the invariance principle to show weak existence of SRBMs under mild conditions. We also use the invariance principle, in conjunction with known uniqueness results for SRBMs, to give some sufficient conditions for validating approximations involving (i) SRBMs in convex polyhedrons with a constant reflection vector field on each face of the polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth boundaries and possibly nonconstant reflection vector fields on the boundary surfaces.
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