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Acta Mathematica Scientia

, Volume 39, Issue 3, pp 645–668 | Cite as

Nonlinear Stochastic Heat Equation Driven by Spatially Colored Noise: Moments and Intermittency

  • Le ChenEmail author
  • Kunwoo KimEmail author
Article
  • 35 Downloads

Abstract

In this article, we study the nonlinear stochastic heat equation in the spatial domain ℝd subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang’s condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on ℤd to that on ℝd. Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [17] using different techniques.

Key words

Stochastic heat equation moment estimates phase transition intermittency intermittency front measure-valued initial data moment Lyapunov exponents 

2010 MR Subject Classification

60H15 35R60 60G60 

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Notes

Acknowledgements

This work was mostly done when both authors were at the University of Utah in the year of 2014. Both authors appreciate many useful comments from and discussions with Davar Khoshnevisan. The first author thanks Daniel Conus for many stimulating discussions on this problem when the first author was invited to give a colloquium talk at Lehigh university in October 2014. The non-symmetric operator “▹” was studied in the first author’s thesis. He would like to express his gratitude to Robert C. Dalang for his careful reading of his thesis.

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Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of NevadaLas VegasUSA
  2. 2.Department of MathematicsPohang University of Science and TechnologyPohang, GyeongbukKorea

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