On the chaotic character of the stochastic heat equation, II

Consider the stochastic heat equation , where the solution u :=  u t ( x ) is indexed by , and is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large- fixed- t behavior of the solution u in different regimes, thereby study the effect of nois...

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Bibliographic Details
Published in:Probability theory and related fields 2013-08, Vol.156 (3-4), p.483-533
Main Authors: Conus, Daniel, Joseph, Mathew, Khoshnevisan, Davar, Shiu, Shang-Yuan
Format: Article
Language:English
Subjects:
Correlation
Economics
Exponents
Finance
Fluctuation
Gaussian
Heat
Heat equations
Insurance
Management
Mathematical analysis
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Noise
Nonlinear equations
Operations Research/Decision Theory
Probability
Probability theory
Probability Theory and Stochastic Processes
Quantitative Finance
Random noise
Spacetime
Statistics for Business
Stochastic models
Stochasticity
Studies
Theoretical
Thermodynamics
Viscosity
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Summary:Consider the stochastic heat equation , where the solution u :=  u t ( x ) is indexed by , and is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large- fixed- t behavior of the solution u in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function f of the noise is of Riesz type, that is , then the “fluctuation exponents” of the solution are for the spatial variable and for the time variable, where . Moreover, these exponent relations hold as long as ; that is precisely when Dalang’s theory [Dalang, Electron J Probab 4:(Paper no. 6):29, 1999 ] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [Kardar et al., Phys Rev Lett 58(20):889–892, 1985 ; Kardar and Zhang, Phys Rev Lett 58(20):2087–2090, 1987 ].
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-012-0434-3