Gillis model, introduced more than 60 years ago, is a non-homogeneous random walk with a position dependent drift. Though parsimoniously cited both in the physical and mathematical literature, it provides one of the very few examples of a stochastic system allowing for a number of exact result, although lacking translational invariance. We present old and novel results for such model, which moreover we show represents a discrete version of a diffusive particle in the presence of a logarithmic potential.

中文翻译：

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探索 Gillis 模型：对数势中扩散的离散方法

60 多年前推出的 Gillis 模型是一种具有位置相关漂移的非均匀随机游走。尽管在物理和数学文献中都被简洁地引用，但它提供了为数不多的随机系统示例之一，尽管缺乏平移不变性，但允许许多精确结果。我们展示了这种模型的新旧结果，此外，我们展示了在对数势存在下扩散粒子的离散版本。