GLOBAL SOLUTIONS TO STOCHASTIC REACTION–DIFFUSION EQUATIONS WITH SUPER-LINEAR DRIFT AND MULTIPLICATIVE NOISE

Robert C. Dalang, Davar Khoshnevisan, Tusheng Zhang

    Research output: Contribution to journalArticlepeer-review

    Abstract

    Let ξ(t,x) denote space–time white noise and consider a reaction–diffusion equation of the form u˙(t,x)=12u"(t,x)+b(u(t,x))+σ(u(t,x))ξ(t,x), on R+×[0,1], with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists ε>0 such that |b(z)|≥|z|(log|z|)1+ε for all sufficiently-large values of |z|. When σ≡0, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [Phys. D 238 (2009) 209–215] have recently shown that there is finite-time blowup when σ is a nonzero constant. In this paper, we prove that the Bonder–Groisman condition is unimprovable by showing that the reaction–diffusion equation with noise is “typically” well posed when |b(z)|=O(|z|log+|z|) as |z|→∞. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.
    Original languageEnglish
    Pages (from-to)519-559
    JournalAnnals of Probability
    Volume47
    Issue number1
    Early online date13 Dec 2018
    DOIs
    Publication statusPublished - 2019

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