关于线性扩散的反演和Doob $h$ -变换
On inversions and Doob $h$-transforms of linear diffusions
摘要：
设$X$是一个正则线性扩散，其状态空间是一个开区间$E\subseteq\mathbb{R}$。我们考虑一个扩散$X^*$，其概率律是$X$的Doob $h$ -变换，其中$h$是$E$上$X$的无穷小生成元的正调和函数.这是$X$相对于$h(x)m(dx)$的对偶，其中$m(dx)$是$X$的速度度量。示例包括$X^*$被调节为保持在某个固定水平以上的$X$的情况。我们提供了一个结构的$X^*$作为一个确定性的反演$X$，时间改变了一些随机时钟。本文的研究涉及到一些反演的构造，这些反演推广了欧几里德反演。详细讨论了带漂移的布朗运动和Bessel过程。
英文摘要：
Let $X$ be a regular linear diffusion whose state space is an open interval $E\subseteq\mathbb{R}$. We consider a diffusion $X^*$ which probability law is obtained as a Doob $h$-transform of the law of $X$, where $h$ is a positive harmonic function for the infinitesimal generator of $X$ on $E$. This is the dual of $X$ with respect to $h(x)m(dx)$ where $m(dx)$ is the speed measure of $X$. Examples include the case where $X^*$ is $X$ conditioned to stay above some fixed level. We provide a construction of $X^*$ as a deterministic inversion of $X$, time changed with some random clock. The study involves the construction of some inversions which generalize the Euclidean inversions. Brownian motion with drift and Bessel processes are considered in details.
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