无界管状区域中随机反射台球的渐近行为
Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains
摘要：
我们研究随机台球在无限平面域的曲线边界：即分段确定性运动的随机性，通过随机反射在域边界。物理动机的过程起源于理想气体模型在克努森制度，粒子反射了微观粗糙的表面。我们将这一过程分为复发性和短暂性两类。我们还给出了几乎肯定的结果的长期行为的位置的粒子，包括超扩散率逃逸的瞬态情况下。在获得我们的结果的一个关键步骤是将我们的过程与一个渐进零漂移的一维随机过程的实例，我们证明了一些新的几乎肯定的独立利益的界限。我们通过一般半鞅准则的应用，也有一些独立的利益，获得这些界限。
英文摘要：
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.
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