Abstract
Consider a stochastic process that behaves as a ddimensional simple and symmetric random walk, except that, with a certain fixed probability, at each step, it chooses instead to jump to a given site with probability proportional to the time it has already spent there. This process has been analyzed in physics literature under the name random walk with preferential relocations, where it is argued that the position of the walker after n steps, scaled by, converges to a Gaussian random variable; because of the spatial scaling, the process is said to undergo a slow diffusion. In this paper, we generalize this model by allowing the underlying random walk to be any Markov process and the random runlengths (time between two relocations) to be i.i.d.distributed. We also allow the memory of the walker to fade with time, meaning that when a relocations occurs, the walker is more likely to go back to a place it has visited more recently. We rigorously prove that the central limit theorem described above (plus a local limit theorem and the convergence of the weighted occupation measure) by associating to the process a growing family of vertexweighted random recursive trees and a Markov chain indexed by this tree. The spatial scaling of our relocated random walk is related to the height of a 'typical' vertex in the random tree. This typical height can range from doublylogarithmic to logarithmic or even a power of the number of nodes of the tree, depending on the form of the memory.
Original language  English 

Article number  093206 
Pages (fromto)  150 
Number of pages  50 
Journal  Journal of Statistical MechanicsTheory and Experiment 
Volume  2019 
Issue number  9 
DOIs  
Publication status  Published  18 Sep 2019 
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Cecile Mailler
 Department of Mathematical Sciences  Lecturer
 Probability Laboratory at Bath
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching, Researcher