论文标题
通过非线性加固的整数上的顶点增强跳跃过程
Vertex-reinforced jump process on the integers with nonlinear reinforcement
论文作者
论文摘要
我们考虑$ \ Mathbb {Z} $上的非线性顶点增强的跳跃过程（VRJP（$ W $）），可测量的重量函数$ w：[1，\ infty）\ to [1，\ infty）$ and [1，\ infty）$，而初始权重等于一个。我们的主要目标是研究VRJP（$ W $）的渐近行为，具体取决于$ w $的倒数。特别是，我们证明，如果$ \ int_1^{\ infty} \ frac {\ text {d} u} {w（u）} = \ infty $，则该过程是经常访问的，即，它无限地访问每个顶点，并且所有本地时代都没有结合。另一方面，如果$ \ int_1^{\ infty} \ frac {\ text {d} u} {w（u）} <\ infty $，并且存在$ρ> 0 $，则$ t \ mapsto w（t）^ρ\ int_t^{\ infty} \ frac {\ text {d} u} {w（u）} $是非犯的，那么该过程最终将恰好粘在三个顶点上，并且只有一个无垃圾时间的顶点。我们还表明，如果初始权重相同，则$ \ mathbb {z} $上的vrjp不能是短暂的，即至少存在一个无限访问的顶点。我们的结果扩展了戴维斯和沃尔科夫先前获得的结果[Probab。理论相关。 Fields（2002）]他表明，在$ \ mathbb {z} $上具有线性增强的VRJP是经常性的。
We consider a non-linear vertex-reinforced jump process (VRJP($w$)) on $\mathbb{Z}$ with an increasing measurable weight function $w:[1,\infty)\to [1,\infty)$ and initial weights equal to one. Our main goal is to study the asymptotic behaviour of VRJP($w$) depending on the integrability of the reciprocal of $w$. In particular, we prove that if $\int_1^{\infty} \frac{\text{d}u}{w(u)} =\infty$ then the process is recurrent, i.e. it visits each vertex infinitely often and all local times are unbounded. On the other hand, if $\int_1^{\infty} \frac{\text{d} u}{w(u)} <\infty$ and there exists a $ρ>0$ such that $t \mapsto w(t)^ρ\int_t^{\infty}\frac{\text{d}u}{w(u)}$ is non-increasing then the process will eventually get stuck on exactly three vertices, and there is only one vertex with unbounded local time. We also show that if the initial weights are all the same, VRJP on $\mathbb{Z}$ cannot be transient, i.e. there exists at least one vertex that is visited infinitely often. Our results extend the ones previously obtained by Davis and Volkov [Probab. Theory Relat. Fields (2002)] who showed that VRJP with linear reinforcement on $\mathbb{Z}$ is recurrent.
