Asymptotic Behaviour of Randomly Reflecting Billiards in Unbounded Tubular Domains
Asymptotic Behaviour of Randomly Reflecting Billiards in Unbounded Tubular Domains
复制标题
无界管域中随机反射台球的渐近行为
DOI：
10.1007/s10955-008-9578-z
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发表时间：
2008
影响因子：
1.6
通讯作者：
A. Wade
中科院分区：
文献类型：
--
作者：
M. Menshikov;M. Vachkovskaia;A. Wade
我们研究具有曲线边界的无限平面域中的随机台球：即通过域边界处的随机反射引入的分段确定性运动和随机性。该过程的物理动机源于克努森体系中的理想气体模型，其中颗粒从微观粗糙表面反射。我们将该过程分为周期性情况和暂时性情况。我们还给出了粒子位置的长期行为的几乎确定的结果，包括瞬态情况下的超扩散逃逸率。获得结果的关键一步是将我们的过程与渐近零漂移的一维随机过程的实例联系起来，为此我们证明了一些新的几乎确定的独立兴趣界限。我们通过应用一般的半鞅标准获得了其中一些界限，这也具有一些独立的意义。
We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle, including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process to an instance of a one-dimensional stochastic process with asymptotically zero drift, for which we prove some new almost-sure bounds of independent interest. We obtain some of these bounds via an application of general semimartingale criteria, also of some independent interest.