15. chen.huang:19:comparison

Comparison principle for stochastic heat equation on \(\mathbb{R}^d\)

Le Chen and Jingyu Huang

Abstract: We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on \(\mathbb{R}^d\)

\[\left(\frac{\partial }{\partial t} -\frac{1}{2}\Delta \right) u(t,x) = \rho(u(t,x)) \:\dot{M}(t,x),\]

for measure-valued initial data, where \(\dot{M}\) is a spatially homogeneous Gaussian noise that is white in time and \(\rho\) is Lipschitz continuous. These results are obtained under the condition that

\[\int_{\mathbb{R}^d}(1+|\xi|^2)^{\alpha-1}\hat{f}(\text{d} \xi)<\infty \quad \text{for some } \alpha\in(0,1],\]

where \(\hat{f}\) is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang’s condition, i.e., \(\alpha=0\). As some intermediate results, we obtain handy upper bounds for \(L^p(\Omega)\)-moments of \(u(t,x)\) for all \(p\ge 2\), and also prove that \(u\) is a.s. Hölder continuous with order \(\alpha-\epsilon\) in space and \(\alpha/2-\epsilon\) in time for any small \(\epsilon>0\).

MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.

Keywords: stochastic heat equation, parabolic Anderson model, space-time Hölder regularity, spatially homogeneous noise, comparison principle, measure-valued initial data.

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[CH19a] Chen, Le & Huang, Jingyu (2019) ‘Comparison principle for stochastic heat equation on \(\mathbb{R}^d\)’, Ann. Probab. 47, 989–1035. <https://doi.org/10.1214/18-AOP1277>

@article{chen.huang:19:comparison,
  author        = {Chen, Le and Huang, Jingyu},
  title         = {Comparison principle for stochastic heat equation on {$\Bbb R^d$}},
  journal       = {Ann. Probab.},
  fjournal      = {The Annals of Probability},
  volume        = {47},
  year          = {2019},
  number        = {2},
  pages         = {989--1035},
  issn          = {0091-1798},
  mrclass       = {60H15 (35B51 35R60 60G60)},
  mrnumber      = {3916940},
  mrreviewer    = {Petru A. Cioica-Licht},
  doi           = {10.1214/18-AOP1277},
  url           = {https://doi.org/10.1214/18-AOP1277}
}

References: Adams and Fournier [AF03b]; Balan and Chen [BC18]; Carmona and Molchanov [CM94]; Chen and Dalang [CD14a]; Chen and Dalang [CD15b]; Chen and Kim [CK17]; Chen and Kim [CK19]; Chen et al. [CHKK19]; Conus et al. [CJK12]; Dalang [Dal99]; Dalang and Quer-Sardanyons [DQS11]; Dawson and Salehi [DS80]; Foondun and Khoshnevisan [FK13]; Gubinelli and Perkowski [GP17]; Hu et al. [HHN16]; Hu et al. [HHNT15]; Huang [Hua17]; Huang et al. [HLN17a]; Moreno Flores [MF14]; Mueller [Mue91b]; Mueller and Nualart [MN08]; Sanz-Solé and Sarrà [SSS02]; Shiga [Shi94]; Tessitore and Zabczyk [TZ98b]; Walsh [Wal86];

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