Hitting times for independent random walks on ℤd

Abstract

We consider a system of asymmetric independent random walks on ℤ^d, denoted by {η_t,t∈ℝ}, stationary under the product Poisson measure ν_ρ of marginal density ρ>0. We fix a pattern $\mathcal{A}$, an increasing local event, and denote by τ the hitting time of $\mathcal{A}$. By using a loss network representation of our system, at small density, we obtain a coupling between the laws of η_t conditioned on {τ>t} for all times t. When d≥3, this provides bounds on the rate of convergence of the law of η_t conditioned on {τ>t} toward its limiting probability measure as t tends to infinity. We also treat the case where the initial measure is close to ν_ρ without being product.