Abstract
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.
Highlights
We consider the stochastic heat equation in R ∂u ∂t =1 2 u + uW, u(0, ·) = u0(·) (1.1)where t ≥ 0, x ∈ R ( ≥ 1) and u0 is a Borel measure
The inverse Fourier transform of μ is in general a distribution defined formally by the expression γ (x) eiξ·x μ(ξ )dξ
Condition (1.6) excludes other initial data of interests such as compactly supported measures. It is our purpose in the current paper to investigate the almost sure spatial asymptotic of the solutions corresponding to these initial data
Summary
Where t ≥ 0, x ∈ R ( ≥ 1) and u0 is a Borel measure. W is a centered Gaussian field, which is white in time and it has a correlated spatial covariance. Suppose for the moment that Wis a space-time white noise and u0 is a function satisfying c ≤ u0(x) ≤ C, for some positive numbers c, C It is first noted in [7] that there exist positive constants c1, c2 such that almost surely c1. Condition (1.6) excludes other initial data of interests such as compactly supported measures It is our purpose in the current paper to investigate the almost sure spatial asymptotic of the solutions corresponding to these initial data. Upon reviewing the method in obtaining (1.8) described previously, one first seeks for an analogous result to (1.10) for general initial data It is noted in [16] that for every u0 satisfying (1.13), one has lim 1 log sup E t→∞ t x ∈R u(t, x) pt ∗ u0(x).
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