arXiv 1202.10012012-02-05math.PR

一类具有局部负漂移的扩散棘轮的渐近性质

Asymptotic properties of certain diffusion ratchets with locally negative drift

Andrej Depperschmidt, Sophia Götz

备注：: 31 pages, 4 figures

摘要：

考虑两个反射扩散过程$(X_t)_{t \ge 0}$，其移动反射边界由非减纯跳马尔可夫过程$(R_t)_{t \ge 0}$给出.在反射边界的跳跃之间，扩散部分表现为具有负漂移的反射布朗运动或反射Ornstein-Uhlenbeck过程。在这两种情况下，在速率$γ(X_t -R_t)$下，对于某些$γ\ge 0$，反射边界跳到在$[R_{t-},X_t]$中均匀选择的新值。由于在反射边界的每一次跳跃之后，扩散在更高的水平上被反射，我们称之为布朗棘轮和奥恩斯坦-乌伦贝克棘轮。这种扩散棘轮是由被动蛋白质跨膜运输的生物学动机。这里考虑的过程是在（Depperschaldom和Pfaffelhuber，2010）中研究的布朗棘轮（无漂移）的概括。对于这两个过程，我们证明了一个法律的大数，特别是每个棘轮移动到无穷大的正速度，可以明确计算，和一个中心极限定理。

英文摘要：

We consider two reflecting diffusion processes $(X_t)_{t \ge 0}$ with a moving reflection boundary given by a non-decreasing pure jump Markov process $(R_t)_{t \ge 0}$. Between the jumps of the reflection boundary the diffusion part behaves as a reflecting Brownian motion with negative drift or as a reflecting Ornstein-Uhlenbeck process. In both cases at rate $γ(X_t -R_t)$ for some $γ\ge 0$ the reflection boundary jumps to a new value chosen uniformly in $[R_{t-},X_t]$. Since after each jump of the reflection boundary the diffusions are reflected at a higher level we call the processes Brownian ratchet and Ornstein-Uhlenbeck ratchet. Such diffusion ratchets are biologically motivated by passive protein transport across membranes. The processes considered here are generalisations of the Brownian ratchet (without drift) studied in (Depperschmidt and Pfaffelhuber, 2010). For both processes we prove a law of large numbers, in particular each of the ratchets moves to infinity at a positive speed which can be computed explicitly, and a central limit theorem.

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