2020/12/23 by Davar Khoshnevisan, Kunwoo Kim, Khoshnevisan, Davar +5 · 1 citation
Computer Science · Mathematics · Medicine · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Primary: 60H15 #Probability (math.PR) #Secondary: 35R60 #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2012.12512
openalex publication_date 2020/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a reaction-diffusion equation of the type ∂tψ= ∂2xψ+ V(ψ) + λσ(ψ)W \texton (0 ,∞)×\mathbbT, subject to a "nice" initial value and periodic boundary, where \mathbbT=[-1 ,1] and W denotes space-time white noise. The reaction term V:ℝ→ℝ belongs to a large family of functions that includes Fisher--KPP nonlinearities [V(x)=x(1-x)] as well as Allen-Cahn potentials [V(x)=x(1-x)(1+x)], the multiplicative nonlinearity σ:ℝ→ℝ is non random and Lipschitz continuous, and λ>0 is a non-random number that measures the strength of the effect of the noise W. The principal finding of this paper is that: (i) When λ is sufficiently large, the above equation has a unique invariant measure; and (ii) When λ is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures.