Nonlinear Stochastic Heat Equation Driven by Spatially Colored Noise: Moments and Intermittency
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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  from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al  using different techniques.
Key wordsStochastic heat equation moment estimates phase transition intermittency intermittency front measure-valued initial data moment Lyapunov exponents
2010 MR Subject Classification60H15 35R60 60G60
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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.
- Carmona René A, Stanislav A Molchanov. Parabolic Anderson problem and intermittency. Mem Amer Math Soc, 1994, 108(518)Google Scholar
- Chen Le. Moments, intermittency, and growth indices for nonlinear stochastic PDE’s with rough initial conditions. PhD Thesis, No 5712. École Polytechnique Fédérale de Lausanne, 2013Google Scholar
- Dalang Robert C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron J Probab, 1999, 4(6): 29 ppGoogle Scholar
- Huang Jingyu. On stochastic heat equation with measure initial data. Electron Commun Probab, 2017, 22(40): 6 ppGoogle Scholar
- Khoshnevisan Davar. Analysis of stochastic partial differential equations. CBMS Regional Conference Series in Mathematics, 119. Published for the Conference Board of the Mathematical Sciences, Washington, DC. Providence, RI: the American Mathematical Society, 2014: viii+116 ppGoogle Scholar
- Podlubny Igor. Fractional differential equations. Volume 198 of Mathematics in Science and Engineering. San Diego, CA: Academic Press Inc, 1999Google Scholar
- Walsh John B. An Introduction to Stochastic Partial Differential Equations//École d’èté de probabilités de Saint-Flour, XIV—1984, 265–439. Lecture Notes in Math 1180. Berlin: Springer, 1986Google Scholar