Summary
In this paper, we consider limit theorems for the asymmetric nearest neighbor exclusion process on the integers. The initial distribution is a product measure with asymptotic density λ at -∞ and ⌕ at +∞. Earlier results described the limiting behavior in all cases except for 0<λ<1/2, λ+⌕=1. Here we treat the exceptional case, which is more delicate. It corresponds to the one in which a shock wave occurs in an associated partial differential equation. In the cases treated earlier, the limit was an extremal invariant measure. By contrast, in the present case the limit is a mixture of two invariant measures. Our theorem resolves a conjecture made by the third author in 1975 [7]. The convergence proof is based on coupling and symmetry considerations.
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Research supported in part by NSF Grant DMS 83-1080
Research supported in part by NSF Grant MCS 83-00836
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Andjel, E.D., Bramson, M.D. & Liggett, T.M. Shocks in the asymmetric exclusion process. Probab. Th. Rel. Fields 78, 231–247 (1988). https://doi.org/10.1007/BF00322020
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DOI: https://doi.org/10.1007/BF00322020