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

## TL;DR

This paper studies the extreme growth behavior of solutions to the stochastic heat equation with localized initial data, revealing how the peaks evolve over large spatial scales under spatially correlated noise.

## Contribution

It extends previous work by analyzing the spatial asymptotics of the stochastic heat equation with compactly supported initial data, including Dirac masses, under correlated noise.

## Key findings

- Growth of peaks characterized over large radii
- Results relate to previous constant initial data studies
- Includes cases with Dirac delta initial conditions

## 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, Joseph and Khoshnevisan 2013 and X. Chen 2016, where constant initial data are considered.