论文标题
自我撤销随机步行的本地时代的波动，有向边缘
Fluctuations of the local times of the self-repelling random walk with directed edges
论文作者
论文摘要
在2008年，Tóth和Vető定义了自我重复的随机步行，有向边缘作为$ \ Mathbb {z} $的非马克维亚随机步行：在此模型中，该步行从$ \ Mathbb {z} $的点移动到给定邻居的概率依赖于本地点的到达端口的范围，该邻居依赖于本地端口的到达端口的数量。他们发现，该模型具有非常特殊的行为，因为所有边缘的当地时间形成的过程在某种类型的停止时间进行了评估，并适当地重新归一化，收敛到确定性过程，而不是类似模型中的随机过程。在这项工作中，我们研究了当地时间过程的波动，围绕其确定性限制，以前尚未知道。我们证明，这些波动在skorohod $ m_1 $拓扑中汇合，以及统一的拓扑结构，远离极限的不连续性，但在最古典的skorohod拓扑中却没有。我们还证明了上述停止时间的波动的融合。
In 2008, Tóth and Vető defined the self-repelling random walk with directed edges as a non-Markovian random walk on $\mathbb{Z}$: in this model, the probability that the walk moves from a point of $\mathbb{Z}$ to a given neighbor depends on the number of previous crossings of the directed edge from the initial point to the target, called the local time of the edge. They found this model had a very peculiar behavior, as the process formed by the local times of all the edges, evaluated at a stopping time of a certain type and suitably renormalized, converges to a deterministic process, instead of a random one as in similar models. In this work, we study the fluctuations of the local times process around its deterministic limit, about which nothing was previously known. We prove that these fluctuations converge in the Skorohod $M_1$ topology, as well as in the uniform topology away from the discontinuities of the limit, but not in the most classical Skorohod topology. We also prove the convergence of the fluctuations of the aforementioned stopping times.
