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• W1997549222 abstract "Consider the stochastic heat equation $${partial_t u = (varkappa/2)Delta u+sigma(u)dot{F}}$$ , where the solution u := u t (x) is indexed by $${(t, x) in (0, infty) times {bf R}^d}$$ , and $${dot{F}}$$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large- $${|x|}$$ 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 $${f(x)propto |x|^{-alpha}}$$ , then the “fluctuation exponents” of the solution are $${psi}$$ for the spatial variable and $${2psi-1}$$ for the time variable, where $${psi:=2/(4-alpha)}$$ . Moreover, these exponent relations hold as long as $${alpha in (0, d wedge 2)}$$ ; 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]." @default.
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• W1997549222 date "2012-05-27" @default.
• W1997549222 modified "2023-10-15" @default.
• W1997549222 title "On the chaotic character of the stochastic heat equation, II" @default.
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• W1997549222 doi "https://doi.org/10.1007/s00440-012-0434-3" @default.
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