Dissipation and high disorder

• Le Chen ^[1] ; Michael Cranston ^[2] ; Davar Khoshnevisan ^[1] ; Kunwoo Kim ^[1]
  1. [1] University of Utah

     University of Utah

     Estados Unidos

     
  2. [2] University of California (Irvine)
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 1, 2017, págs. 82-99
• Idioma: inglés
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• Resumen

  • Given a field {B(x)}x∈Zd{B(x)}x∈Zd of independent standard Brownian motions, indexed by ZdZd, the generator of a suitable Markov process on Zd,GZd,G, and sufficiently nice function σ:[0,∞)↦[0,∞)σ:[0,∞)↦[0,∞), we consider the influence of the parameter λλ on the behavior of the system, dut(x)u0(x)==(Gut)(x)dt+λσ(ut(x))dBt(x)[t>0, x∈Zd],c0δ0(x).

    dut(x)=(Gut)(x)dt+λσ(ut(x))dBt(x)[t>0, x∈Zd],u0(x)=c0δ0(x).

    We show that for any λ>0λ>0 in dimensions one and two the total mass ∑x∈Zdut(x)∑x∈Zdut(x) converges to zero as t→∞t→∞ while for dimensions greater than two there is a phase transition point λc∈(0,∞)λc∈(0,∞) such that for λ>λc,∑x∈Zdut(x)→0λ>λc,∑x∈Zdut(x)→0 as t→∞t→∞ while for λ<λc,∑x∈Zdut(x)↛0λ<λc,∑x∈Zdut(x)↛0 as t→∞t→∞.