具有广义时空高斯噪声的抛物型Anderson模型的空间渐近性

IF 2.1 1区数学 Q1 STATISTICS & PROBABILITY

Annals of Probability Pub Date : 2016-03-01 DOI:10.1214/15-AOP1006

Xia Chen

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引用次数: 44

摘要

部分原因是Conus, Joseph和Khoshnevisan最近的论文[Ann]。[Probab. 41(2013) 2225-2260]和Conus等人。理论相关领域156(2013)483-533]，本文研究了抛物型Anderson方程{∂u∂t(t,x)=12Δu(t,x)+V(t,x)u(t,x)，u(0,x)=u0(x)的精确空间渐近行为，其中齐次广义高斯噪声V(t,x)在时间和空间上为白色或分数白色。结合KPZ方程的Cole-Hopf解，得到了精确的渐近形式limR→∞(logR)−2/3logmax|x|≤Ru(t,x)=342t3−−−√3a。为抛物线型安德森模型∂tu=12∂2xxu+W˙u，具有(1+1)-白噪声W˙(t,x)。此外，还讨论了抛物型安德森方程的时间渐近性与空间渐近性之间的联系。

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Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise

Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225–2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483–533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation {∂u∂t(t,x)=12Δu(t,x)+V(t,x)u(t,x),u(0,x)=u0(x), where the homogeneous generalized Gaussian noise V(t,x) is, among other forms, white or fractional white in time and space. Associated with the Cole–Hopf solution to the KPZ equation, in particular, the precise asymptotic form limR→∞(logR)−2/3logmax|x|≤Ru(t,x)=342t3−−−√3a.s. is obtained for the parabolic Anderson model ∂tu=12∂2xxu+W˙u with the (1+1)-white noise W˙(t,x). In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.

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来源期刊

Annals of Probability 数学-统计学与概率论

CiteScore

4.60

自引率

8.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The Annals of Probability publishes research papers in modern probability theory, its relations to other areas of mathematics, and its applications in the physical and biological sciences. Emphasis is on importance, interest, and originality – formal novelty and correctness are not sufficient for publication. The Annals will also publish authoritative review papers and surveys of areas in vigorous development.