SPATIAL ASYMPTOTICS FOR THE PARABOLIC ANDERSON MODELS WITH GENERALIZED TIME-SPACE GAUSSIAN NOISE

• 成果类型: Article
• 署名作者: Chen, Xia
• 署名单位: University of Tennessee System; University of Tennessee Knoxville
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1006
• 发表日期: 2016
• 页码: 1535-1598
• 关键词: stochastic heat-equation; Feynman-kac Formula; exponential asymptotics; chaotic character; local-times; driven

摘要

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 {partial derivative u/partial derivative t (t, x) = 1/2 Delta u(t, x) + V (t, x)u(t, x), u(0, x) = u(0)(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 ColeHopf solution to the KPZ equation, in particular, the precise asymptotic form lim R ->infinity (logR)(-2/3) log max(vertical bar x vertical bar <= R) u(t, x) = 3/4 3 root 2t/3 a.s. is obtained for the parabolic Anderson model partial derivative(t)u = 1/2 partial derivative(2)(xx)u + (W) over dotu with the (1 + 1)-white noise (W) over dot(t, x). In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.