Projects per year
Abstract
We consider a nonlinear vertexreinforced jump process (VRJP(w)) on Z with an increasing measurable weight function w : [1, ∞) → [1, ∞) 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 (Equation presented). then the process is recurrent, that is, it visits each vertex infinitely often and all local times are unbounded. On the other hand, if (Equation presented). and there exists a ρ > 0 such that (Equation presented). du is nonincreasing then the process will eventually get stuck on exactly three w(u) 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, 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) 281300) who showed that VRJP with linear reinforcement on ℤ is recurrent.
Original language  English 

Pages (fromto)  26712705 
Number of pages  35 
Journal  Annals of Applied Probability 
Volume  32 
Issue number  4 
DOIs  
Publication status  Published  Aug 2022 
Keywords
 localization
 random processes with reinforcement
 Selfinteracting processes
 vertexreinforced jump processes
Projects
 2 Finished

Random walks with long memory
Collevecchio, A., Garoni, T., Hamza, K., Cotar, C. & Tarres, P.
Australian Research Council (ARC)
1/05/18 → 1/05/22
Project: Research

ARC Centre of Excellence for Mathematical and Statistical Frontiers of Big Data, Big Models, New Insights
Hall, P., Bartlett, P., Bean, N., Burrage, K., DeGier, J., Delaigle, A., Forrester, P., Geweke, J., Kohn, R., Kroese, D., Mengersen, K. L., Pettit, A., Pollett, P., Roughan, M., Ryan, L. M., Taylor, P., Turner, I., Wand, M., Garoni, T., SmithMiles, K. A., Caley, M., Churches, T., Elazar, D., Gupta, A., Harch, B., Tam, S., Weegberg, K., Willinger, W. & Hyndman, R.
Australian Research Council (ARC), Monash University – Internal Department Contribution, University of Melbourne, Queensland University of Technology (QUT), University of Adelaide, University of New South Wales (UNSW), University of Queensland , University of Technology (UTS) Sydney, Monash University – Internal University Contribution, Monash University – Internal Faculty Contribution, Monash University – Internal School Contribution, Roads Corporation (trading as VicRoads) (Victoria)
1/01/17 → 31/12/21
Project: Research