%0 Generic %1 collevecchio2020vertexreinforced %A Collevecchio, Andrea %A Nguyen, Tuan-Minh %A Volkov, Stanislav %D 2020 %K Probability %T Vertex-reinforced jump process on the integers with nonlinear reinforcement %U http://arxiv.org/abs/2004.05927 %X We consider non-linear vertex reinforced jump process (VRJP($w$)) on $Z$ with an increasing measurable weight function $w:1,ınfty)\to 1,ınfty)$ 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 $1/w L^1(1,ınfty), Leb)$ then the process is recurrent, i.e. it visits each vertex infinitely often and all local times are unbounded. On the other hand, if $1/w L^1(1,ınfty), Leb)$ and there exists a $\rho>0$ such that $t w(t)^\rhoınt_t^ınftyduw(u)$ 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 $Z$ 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 $Z$ is recurrent.

@preprint{collevecchio2020vertexreinforced, abstract = {We consider 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 $1/w \notin L^1([1,\infty), {\rm Leb})$ then the process is recurrent, i.e. it visits each vertex infinitely often and all local times are unbounded. On the other hand, if $1/w \in L^1([1,\infty), {\rm Leb})$ and there exists a $\rho>0$ such that $t \mapsto w(t)^{\rho}\int_t^{\infty}\frac{du}{w(u)}$ 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.}, added-at = {2020-04-14T04:55:43.000+0200}, author = {Collevecchio, Andrea and Nguyen, Tuan-Minh and Volkov, Stanislav}, biburl = {https://www.bibsonomy.org/bibtex/29ca00f7b5d6cbb82555df4180b681410/vnkvant}, description = {Vertex-reinforced jump process on the integers with nonlinear reinforcement}, interhash = {08917f5b9226f2f0e03f7c5a6d94e6c6}, intrahash = {9ca00f7b5d6cbb82555df4180b681410}, keywords = {Probability}, note = {cite arxiv:2004.05927Comment: 38 pages}, timestamp = {2020-04-14T04:55:43.000+0200}, title = {Vertex-reinforced jump process on the integers with nonlinear reinforcement}, url = {http://arxiv.org/abs/2004.05927}, year = 2020 }