A random walk on Z with drift driven by its occupation time at zero

2007/11/30 by Iddo Ben-Ari, Ben-Ari, Iddo, Mathieu Merle +3
Mathematics · #60F17 #60F20 #60F5 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F17 #msc:60F20 #msc:60F5

paper · pdf · doi:10.48550/arxiv.0711.4871

arxiv created 2007/11/30 · arxiv updated 2009/12/01

Abstract

We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes according to the rate of decay of the drift. In particular, when the rate is sufficiently slow, the position of the random walk, properly normalized, converges to a symmetric exponential law. In this regime, in contrast to the classical case, the range of the walk scales differently from its position.

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