基于广义梯高函数的多维有序随机游动
Multidimensional random walks conditioned to stay ordered via generalized ladder height functions
摘要：
条件为正的随机游动是涨落理论中的一个重要课题。构造它们的一种方法是随机游动，条件是在时间$n$之前保持正值，并让$n$趋于无穷大。第二种方法是条件反射，直到一个独立的几何时间保持正值，并将其参数发送到零。多维情况（条件是$d$维随机游动的分量是有序的）在[EK 08]中使用第一种方法求解，但需要施加一些矩条件。我们的方法是基于第二种方法，它的优点是需要一个最小的限制，只需要在某些情况下的有限性的$h$变换。我们还刻画了极限是马氏极限或次马氏极限的性质，并给出了$h$ -函数的几种重新表示.在文献[18]给出的条件下，可以证明我们的$h$ -函数是唯一在Weyl腔外为零的调和函数$\{x=(x_1,\ldots, x_d)\in \mathbb{R}^d: x_1<\cdots < x_d\}$ .
英文摘要：
Random walks conditioned to stay positive are a prominent topic in fluctuation theory. One way to construct them is as a random walk conditioned to stay positive up to time $n$, and let $n$ tend to infinity. A second method is conditioning instead to stay positive up to an independent geometric time, and send its parameter to zero. The multidimensional case (condition the components of a $d$-dimensional random walk to be ordered) was solved in [EK08] using the first approach, but some moment conditions need to be imposed. Our approach is based on the second method, which has the advantage to require a minimal restriction, needed only for the finiteness of the $h$-transform in certain cases. We also characterize when the limit is Markovian or sub-Markovian, and give several reexpresions of the $h$-function. Under some conditions given in [Ign18], it can be proved that our $h$-function is the only harmonic function which is zero outside the Weyl chamber $\{x=(x_1,\ldots, x_d)\in \mathbb{R}^d: x_1<\cdots < x_d\}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
