Asymptotic properties of certain diffusion ratchets with locally negative drift

TL;DR

This paper studies two types of diffusion ratchets with moving reflection boundaries, proving they move to infinity at positive speeds and establishing their asymptotic behavior, motivated by biological protein transport.

Contribution

It extends previous models by analyzing diffusion ratchets with negative drift, providing explicit speed calculations and asymptotic properties.

Findings

Both ratchets move to infinity at positive speeds.

Explicit formulas for the speeds of the ratchets.

Central limit theorems established for the processes.

Abstract

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 $\gamma(X_t -R_t)$ for some $\gamma \ge 0$ 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.