Spatial asymptotic of the stochastic heat equation with compactly supported initial data

Jingyu Huang, Khoa Lê

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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.
Original languageEnglish
Pages (from-to)495-539
Number of pages45
JournalStochastics and Partial Differential Equations: Analysis and Computations
Volume7
Issue number3
Early online date2 Mar 2019
DOIs
Publication statusPublished - 15 Sep 2019

Keywords

  • Parabolic Anderson model
  • Feynman–Kac representation ·
  • Brownian motion
  • Spatial asymptotic

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