Vertex-reinforced jump process on the integers with nonlinear reinforcement

Abstract

We consider a nonlinear vertex-reinforced jump process (VRJP(w)) on $\mathbb{Z}$ with an increasing measurable weight function $\mathit{w}:[1,\infty )\to [1,\infty )$ 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 ${\int }_1^{\infty }\frac{\mathrm{d}\mathit{u}}{\mathit{w}(\mathit{u})}=\infty$ then the process is recurrent, that is, it visits each vertex infinitely often and all local times are unbounded. On the other hand, if ${\int }_1^{\infty }\frac{\mathrm{d}\mathit{u}}{\mathit{w}(\mathit{u})}<\infty$ and there exists a $\mathit{\rho }>0$ such that $\mathit{t}\mapsto \mathit{w}{(\mathit{t})}^{\mathit{\rho }}{\int }_{\mathit{t}}^{\infty }\frac{\mathrm{d}\mathit{u}}{\mathit{w}(\mathit{u})}$ is nonincreasing 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, that is, there exists at least one vertex that is visited infinitely often. Our results extend the ones previously obtained by Davis and Volkov (Probab. Theory Related Fields 123 (2002) 281–300) who showed that VRJP with linear reinforcement on $\mathbb{Z}$ is recurrent.