On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries
On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries
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关于分段平滑边界域中反射扩散的下鞅问题
DOI：
10.1214/16-aop1153
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
K. Ramanan
中科院分区：
文献类型：
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作者：
W. Kang;K. Ramanan
两个框架已被用来表征反射扩散包括随机微分方程的反射和所谓的下鞅问题。我们引入了具有分段C^2边界和分段连续反射向量场的区域中（斜）反射扩散的下鞅问题的一般形式。在适当的假设下，证明了下鞅问题的适定性等价于相应的带反射的随机微分方程弱解的存在唯一律.我们的结果推广了Stroock和Varadhan关于d维欧氏空间中随机微分方程弱解的适定性与鞅问题适定性的等价性的经典结果。在具有非光滑边界的域中反射扩散的情况下的分析要微妙得多，并且需要仔细分析域边界上反射扩散的行为。特别是，当我们的假设不满足时，等价性可能不成立。我们建立的等价性允许一个转移的反射扩散的结果，其特征在于一种方法，反射扩散分析的其他方法。作为一个应用，我们提供了一个大类的反射扩散的凸多面体区域的平稳分布的特征。
Two frameworks that have been used to characterize reflected diffusions include stochastic differential equations with reflection and the so-called submartingale problem. We introduce a general formulation of the submartingale problem for (obliquely) reflected diffusions in domains with piecewise C^2 boundaries and piecewise continuous reflection vector fields. Under suitable assumptions, we show that well-posedness of the submartingale problem is equivalent to existence and uniqueness in law of weak solutions to the corresponding stochastic differential equation with reflection. Our result generalizes to the case of reflecting diffusions a classical result due to Stroock and Varadhan on the equivalence of well-posedness of martingale problems and well-posedness of weak solutions of stochastic differential equations in d-dimensional Euclidean space. The analysis in the case of reflected diffusions in domains with non-smooth boundaries is considerably more subtle and requires a careful analysis of the behavior of the reflected diffusion on the boundary of the domain. In particular, the equivalence can fail to hold when our assumptions are not satisfied. The equivalence we establish allows one to transfer results on reflected diffusions characterized by one approach to reflected diffusions analyzed by the other approach. As an application, we provide a characterization of stationary distributions of a large class of reflected diffusions in convex polyhedral domains.