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On the chaotic character of the stochastic heat equation, II

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 \({2\psi-1}\) 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].

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Correspondence to Shang-Yuan Shiu.

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This research was supported in part by the NSFs grant DMS-0747758 (M.J.) and DMS-1006903 (D.K.).

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Conus, D., Joseph, M., Khoshnevisan, D. et al. On the chaotic character of the stochastic heat equation, II. Probab. Theory Relat. Fields 156, 483–533 (2013). https://doi.org/10.1007/s00440-012-0434-3

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Keywords

  • The stochastic heat equation
  • Chaos
  • Intermittency
  • The parabolic Anderson model
  • The KPZ equation
  • Critical exponents

Mathematics Subject Classification

  • Primary 60H15
  • Secondary 35R60