Asymptotic expansion of the invariant measure for ballistic random walk in the low disorder regime

• Alejandro F. Ramírez ^[1] ; David Campos
  1. [1] Pontifica Universidad Católica de Chile (Chile)
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 6, 2, 2017, págs. 4675-4699
• Idioma: inglés
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• Resumen

  • We consider a random walk in random environment in the low disorder regime on ZdZd, that is, the probability that the random walk jumps from a site xx to a nearest neighboring site x+ex+e is given by p(e)+εξ(x,e)p(e)+εξ(x,e), where p(e)p(e) is deterministic, {{ξ(x,e):|e|1=1}:x∈Zd}{{ξ(x,e):|e|1=1}:x∈Zd} are i.i.d. and ε>0ε>0 is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in εε for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in εε for the invariant measure of random perturbations of the simple symmetric random walk in dimensions d=2d=2.