18. chen.kim:20:stochastic

Stochastic comparisons for stochastic heat equation

Le Chen and Kunwoo Kim

Abstract: We establish the stochastic comparison principles, including moment comparison principle as a special case, for solutions to the following nonlinear stochastic heat equation on \(\mathbb{R}^d\)

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

where \(\dot{M}\) is a spatially homogeneous Gaussian noise that is white in time and colored in space, and \(\rho\) is a Lipschitz continuous function that vanishes at zero. These results are obtained for rough initial data and under Dalang’s condition, namely,

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

where \(\hat{f}\) is the spectral measure of the noise. We first show that the nonlinear stochastic heat equation can be approximated by systems of interacting diffusions (SDEs) and then, using those approximations, we establish the comparison principles by comparing either the diffusion coefficient \(\rho\) or the correlation function of the noise \(f\). As corollaries, we obtain Slepian’s inequality for SPDEs and SDEs.

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

Keywords: stochastic heat equation, parabolic Anderson model, infinite dimensional SDE, spatially homogeneous noise, stochastic comparison principle, moment comparison principle, Slepian’s inequality for SPDEs, rough initial data.

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[CK20] Chen, Le & Kim, Kunwoo (2020) ‘Stochastic comparisons for stochastic heat equation’, Electron. J. Probab. 25, Paper No. 140, 38. <https://doi.org/10.1214/20-ejp541>

@article{chen.kim:20:stochastic,
  author        = {Chen, Le and Kim, Kunwoo},
  title         = {Stochastic comparisons for stochastic heat equation},
  journal       = {Electron. J. Probab.},
  fjournal      = {Electronic Journal of Probability},
  volume        = {25},
  year          = {2020},
  pages         = {Paper No. 140, 38},
  mrclass       = {60H15 (35R60 60G60)},
  mrnumber      = {4186259},
  mrreviewer    = {Vitalii Konarovskyi},
  doi           = {10.1214/20-ejp541},
  url           = {https://doi.org/10.1214/20-ejp541}
}

References: Balan and Chen [BC18]; Borodin and Corwin [BC14c]; Carmona and Molchanov [CM94]; Chen and Dalang [CD15b]; Chen and Kim [CK17]; Chen and Kim [CK19]; Chen [Che15]; Chen [Che19]; Cox et al. [CFG96]; Dalang [Dal99]; Ethier and Kurtz [EK86]; Foondun et al. [FJL18]; Foondun and Khoshnevisan [FK13]; Friedman [Fri75]; Geiß and Manthey [GM94]; Huang [Hua17]; Ikeda and Watanabe [IW89]; Joseph et al. [JKM17]; Khoshnevisan et al. [KKX17]; Khoshnevisan et al. [KKX18]; Kim [Kim19]; Mueller [Mue91b]; Revuz and Yor [RY99]; Shiga [Shi94]; Shiga and Shimizu [SS80]; Walsh [Wal86];

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