NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY

doi:10.1007/s10473-019-0303-6

Abstract

Abstract: In this article, we study the nonlinear stochastic heat equation in the spatial domain R^d 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 Z^d to that on R^d. 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

CLC Number:

60H15

Le CHEN, Kunwoo KIM. NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY[J].Acta mathematica scientia,Series B, 2019, 39(3): 645-668.

Trendmd

References

[1] Balan Raluca M, Le Chen. Parabolic Anderson model with space-time homogeneous Gaussian noise and rough initial condition. J Theoret Probab, 2018, 31(4):2216-2265
[2] Carmona René A, Stanislav A Molchanov. Parabolic Anderson problem and intermittency. Mem Amer Math Soc, 1994, 108(518)
[3] 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, 2013
[4] Chen Le, Robert C Dalang. Moments and growth indices for nonlinear stochastic heat equation with rough initial conditions. Ann Probab, 2015, 43(6):3006-3051
[5] Chen Le, Jingyu Huang. Comparison principle for stochastic heat equation on R^d. Ann Probab, 2019, 47(2):989-1035. arXiv:1607.03998(2016)
[6] Chen Le, Yaozhong Hu, David Nualart. Two-point correlation function and Feynman-Kac formula for the stochastic heat equation. Potential Anal, 2017, 46(4):779-793
[7] Chen Le, Kunwoo Kim. On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations. Ann Inst Henri Poincar Probab Stat, 2017, 53(1):358-388
[8] Chen Xia. Moment asymptotics for parabolic Anderson equation with fractional time-space noise:Skorokhod regime. Ann Inst Henri Poincar Probab Stat, 2017, 53(2):819-841
[9] Conus Daniel, Davar Khoshnevisan. On the existence and position of the farthest peaks of a family of stochastic heat and wave equations. Probab Theory Related Fields, 2012, 152(3/4):681-701
[10] 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 pp
[11] Dalang Robert C, Nicholas E Frangos. The stochastic wave equation in two spatial dimensions. Ann Probab, 1998, 26(1):187-212
[12] Foondun Mohammud, Davar Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations. Electron J Probab, 2009, 14(21):548-568
[13] Foondun Mohammud, Davar Khoshnevisan. On the stochastic heat equation with spatially-colored random forcing. Trans Amer Math Soc, 2013, 365(1):409-458
[14] Foondun Mohammud, Davar Khoshnevisan. Corrections and improvements to:"On the stochastic heat equation with spatially-colored random forcing". Trans Amer Math Soc, 2014, 366(1):561-562
[15] Foondun Mohammud, Wei Liu, McSylvester Omaba. Moment bounds for a class of fractional stochastic heat equations. Ann Probab, 2017, 45(4):2131-2153
[16] Huang Jingyu. On stochastic heat equation with measure initial data. Electron Commun Probab, 2017, 22(40):6 pp
[17] Huang Jingyu, Khoa Lê, David Nualart. Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise. Ann Inst Henri Poincaré Probab Stat, 2017, 53(3):1305-1340
[18] 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 pp
[19] Khoshnevisan Davar, Kunwoo Kim. Non-linear noise excitation of intermittent stochastic PDEs and the topology of LCA groups. Ann Probab, 2015, 43(4):1944-1991
[20] Mueller Carl. On the support of solutions to the heat equation with noise. Stoch & Stoch Rep, 1991, 37(4):225-245
[21] Noble John M. Evolution equation with Gaussian potential. Nonlinear Anal, 1997, 28(1):103-135
[22] Podlubny Igor. Fractional differential equations. Volume 198 of Mathematics in Science and Engineering. San Diego, CA:Academic Press Inc, 1999
[23] Stein Elias M. Singular integrals and differentiability properties of functions. Princeton, NJ:Princeton University Press, 1970
[24] Tessitore, Gianmario, Jerzy Zabczyk. Strict positivity for stochastic heat equations. Stochastic Process Appl, 1998, 77(1):83-98
[25] Walsh John B. An Introduction to Stochastic Partial Differential Equations//Ècole d'èté de probabilités de Saint-Flour, XIV-1984, 265-439. Lecture Notes in Math 1180. Berlin:Springer, 1986