论文标题
随机反应扩散方程家族的相位分析
Phase Analysis for a family of Stochastic Reaction-Diffusion Equations
论文作者
论文摘要
我们认为类型的反应扩散方程\ [ \partial_tψ= \ partial^2_xψ + v（ψ） +λσ（ψ）\ dot {w} \qquad\text{on $(0\,,\infty)\times\mathbb{T}$}, \] subject to a "nice" initial value and periodic boundary, where $\mathbb{T}=[-1\,,1]$ and $\dot{W}$ denotes space-time white noise.反应术语$ v：\ mathbb {r} \ to \ mathbb {r} $属于包括Fisher-kpp非线性在内的大型函数家族[$ v（x）= x（1-x）$]以及allen-cahn电位[$ v（x）= x（x）= x（1+x）（1+x）（1+x）， $σ：\ mathbb {r} \ to \ mathbb {r} $是非随机的，Lipschitz的连续，$λ> 0 $是一个非随机数，可衡量噪声$ \ dot {w} $的效果的强度。 本文的主要发现是：（i）当$λ$足够大时，上述方程式具有独特的不变度； （ii）当$λ$足够小时，所有不变措施的收集是一个非平凡的线段，特别是无限的。这证明了Zimmerman等人的早期预测。 （2000）。我们的方法还对这些不变措施的结构说了很多。
We consider a reaction-diffusion equation of the type \[ \partial_tψ= \partial^2_xψ+ V(ψ) + λσ(ψ)\dot{W} \qquad\text{on $(0\,,\infty)\times\mathbb{T}$}, \] subject to a "nice" initial value and periodic boundary, where $\mathbb{T}=[-1\,,1]$ and $\dot{W}$ denotes space-time white noise. The reaction term $V:\mathbb{R}\to\mathbb{R}$ 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 $σ:\mathbb{R}\to\mathbb{R}$ is non random and Lipschitz continuous, and $λ>0$ is a non-random number that measures the strength of the effect of the noise $\dot{W}$. 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.
