论文标题
莱维随机媒体随机散步的梯子成本
Ladder costs for random walks in Lévy random media
论文作者
论文摘要
我们考虑在Lévy随机培养基上移动的随机步行$ y $，即一个一维续签点过程，在吸引稳定定律的范围内的点之间具有相互距离。重点是对第一级高度$ y _ _ {\ Mathcal {t}} $和长度$ l _ {\ Mathcal {t}}（y）$的定律的特征，其中$ \ nathcal {t} $是$ y $ in $ y $ in $ y $ y n $ \ mathbbbbbbbbbbbbbbb {该研究依赖于我们简要描述的纽带（RWRSB）的随机风景（RWRSB）中的随机风光的构建，表示随机步行。该风景是通过将两个随机变量与$ \ Mathbb {z} $的每个键关联的，与该债券的两个可能的交叉说明相对应。 I.I.D增量的$ \ Mathbb {z} $上的随机步行$ s $收集了它穿过的债券的风景值：我们表示此复合过程RWRSB。在合适的假设下，我们表征了收集到第一个退出时间$ \ MATHCAL {T} $的尾部分布。此设置将用于获得第一级长度和高度$ y $的法律的结果。调查的主要工具是我们沿证明的广义spitzer-baxter身份，以及根据随机步行$ s $的当地时代的适当表示RWRSB。所有这些结果很容易概括为梯子变量的整个序列。
We consider a random walk $Y$ moving on a Lévy random medium, namely a one-dimensional renewal point process with inter-distances between points that are in the domain of attraction of a stable law. The focus is on the characterization of the law of the first-ladder height $Y_{\mathcal{T}}$ and length $L_{\mathcal{T}}(Y)$, where $\mathcal{T}$ is the first-passage time of $Y$ in $\mathbb{R}^+$. The study relies on the construction of a broader class of processes, denoted Random Walks in Random Scenery on Bonds (RWRSB) that we briefly describe. The scenery is constructed by associating two random variables with each bond of $\mathbb{Z}$, corresponding to the two possible crossing directions of that bond. A random walk $S$ on $\mathbb{Z}$ with i.i.d increments collects the scenery values of the bond it traverses: we denote this composite process the RWRSB. Under suitable assumptions, we characterize the tail distribution of the sum of scenery values collected up to the first exit time $\mathcal{T}$. This setting will be applied to obtain results for the laws of the first-ladder length and height of $Y$. The main tools of investigation are a generalized Spitzer-Baxter identity, that we derive along the proof, and a suitable representation of the RWRSB in terms of local times of the random walk $S$. All these results are easily generalized to the entire sequence of ladder variables.
