We investigate the growth of the tallest peaks of random field solutions to the parabolic Anderson models over concentric balls as the radii approach infinity. The noise is white in time and correlated in space. The spatial correlation function is either bounded or non-negative satisfying Dalang’s condition. The initial data are Borel measures with compact supports, in particular, include Dirac masses. The results obtained are related to those of Conus et al. (Ann Probab 41(3B):2225–2260, 2013) and Chen (Ann Probab 44(2):1535–1598, 2016) where constant initial data are considered.

中文翻译：

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具有紧密支持的初始数据的随机热方程的空间渐近性

当半径接近无穷大时，我们研究同心球上抛物线安德森模型的随机场解的最高峰的增长。噪声在时间上是白色的，在空间上是相关的。空间相关函数是满足大朗条件的有界或非负函数。初始数据是带有紧凑支撑的Borel量度，尤其是Dirac质量。获得的结果与Conus等人的结果有关。（Ann Probab 41（3B）：2225-2260，2013）和Chen（Ann Probab 44（2）：1535-1598，2016），其中考虑了恒定的初始数据。