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

2015/11/09 by David Campos, Campos, David, Alejandro F. Ramı́rez +1
Mathematics · Physics and Astronomy · #60K37 #82C41 #82D30 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1511.02945

openalex publication_date 2015/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a random walk in random environment in the low disorder regime on\n mathbb Z^d. That is, the probability that the random walk jumps from a site\nx to a nearest neighboring site x+e is given by p(e)+\ε \ξ(x,e),\nwhere p(e) is deterministic, \ξ(x,e):|e|₁=1 :x\∈ mathbb Z^d are\ni.i.d. and \ε>0 is a parameter which is eventually chosen small enough.\nWe establish an asymptotic expansion in \ε for the invariant measure of\nthe environmental process whenever a ballisticity condition is satisfied. As an\napplication of our expansion, we derive a numerical expression up to first\norder in \ε for the invariant measure of random perturbations of the\nsimple symmetric random walk in dimensions d=2.\n

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