The Annals of Probability

On the chaotic character of the stochastic heat equation, before the onset of intermitttency

Daniel Conus, Mathew Joseph, and Davar Khoshnevisan

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We consider a nonlinear stochastic heat equation $\partial_{t}u=\frac{1}{2}\partial_{xx}u+\sigma(u)\partial_{xt}W$, where $\partial_{xt}W$ denotes space–time white noise and $\sigma:\mathbf{R} \to\mathbf{R} $ is Lipschitz continuous. We establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_{0}$: under suitable conditions on $u_{0}$ and $\sigma$, $\sup_{x\in\mathbf{R} }u_{t}(x)$ is a.s. finite when $u_{0}$ has compact support, whereas with probability one, $\limsup_{|x|\to\infty}u_{t}(x)/({\log}|x|)^{1/6}>0$ when $u_{0}$ is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.

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Ann. Probab., Volume 41, Number 3B (2013), 2225-2260.

First available in Project Euclid: 15 May 2013

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Stochastic heat equation chaos intermittency


Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar. On the chaotic character of the stochastic heat equation, before the onset of intermitttency. Ann. Probab. 41 (2013), no. 3B, 2225--2260. doi:10.1214/11-AOP717.

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  • [1] Balázs, M., Quastel, J. and Seppäläinen, T. (2011). Fluctuation exponent of the KPZ/stochastic Burgers equation. J. Amer. Math. Soc. 24 683–708.
  • [2] Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78 1377–1401.
  • [3] Burkholder, D. L. (1966). Martingale transforms. Ann. Math. Statist. 37 1494–1504.
  • [4] Burkholder, D. L., Davis, B. J. and Gundy, R. F. (1972). Integral inequalities for convex functions of operators on martingales. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory 223–240. Univ. California Press, Berkeley, CA.
  • [5] Burkholder, D. L. and Gundy, R. F. (1970). Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124 249–304.
  • [6] Carlen, E. and Krée, P. (1991). $L^{p}$ estimates on iterated stochastic integrals. Ann. Probab. 19 354–368.
  • [7] Carmona, R. A. and Molchanov, S. A. (1994). Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 viii+125.
  • [8] Chertkov, M., Falkovich, G., Kolokolov, I. and Lebedev, V. (1995). Statistics of a passive scalar advected by a large-scale two-dimensional velocity field: Analytic solution. Phys. Rev. E (3) 51 5609–5627.
  • [9] Conus, D. and Khoshnevisan, D. (2010). Weak nonmild solutions to some SPDEs. Preprint. Available at
  • [10] Dalang, R., Khoshnevisan, D., Mueller, C., Nualart, D. and Xiao, Y. (2009). A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math. 1962. Springer, Berlin.
  • [11] Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 29 pp. (electronic).
  • [12] Davis, B. (1976). On the $L^{p}$ norms of stochastic integrals and other martingales. Duke Math. J. 43 697–704.
  • [13] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B. Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam. Translated from the French by J. P. Wilson.
  • [14] Durrett, R. (1998). Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks Cole, Pacific Grove, CA.
  • [15] Foondun, M. and Khoshnevisan, D. (2009). Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 548–568.
  • [16] Foondun, M. and Khoshnevisan, D. (2010). On the global maximum of the solution to a stochastic heat equation with compact-support initial data. Ann. Inst. Henri Poincaré Probab. Stat. 46 895–907.
  • [17] Gel’fand, I. M. and Vilenkin, N. Y. (1964). Generalized Functions. Vol. 4: Applications of Harmonic Analysis. Academic Press, New York.
  • [18] Kardar, M., Parisi, G. and Zhang, Y. C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889–892.
  • [19] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • [20] Molchanov, S. A. (1991). Ideas in the theory of random media. Acta Appl. Math. 22 139–282.
  • [21] Mueller, C. (1991). On the support of solutions to the heat equation with noise. Stochastics Stochastics Rep. 37 225–245.
  • [22] Ruelle, D. (1985). The onset of turbulence: A mathematical introduction. In Turbulence, Geophysical Flows, Predictability and Climate Dynamics (“Turbolenza e Predicibilità nella Fluidodinamica Geofisica e la Dinamica del Clima,” Rendiconti della Scuola Internazionale di Fisica “Enrico Fermi”). North-Holland, Amsterdam.
  • [23] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.
  • [24] Zel’dovich, Y. B., Molchanov, S. A., Ruzmaĭkin, A. A. and Sokoloff, D. D. (1988). Intermittency, diffusion, and generation in a nonstationary random medium. Sov. Sci. Rev. C Math. Phys. 7 1–110.
  • [25] Zeldovich, Y. B., Molchanov, S. A., Ruzmaikin, A. A. and Sokolov, D. D. (1985). Intermittency of passive fields in random media. J. Exp. Theor. Phys. 89 2061–2072. [Actual journal title: Zhurnal eksperimentalnoi teoreticheskoi fiziki.] (In Russian.)
  • [26] Zel’dovich, Y. B., Ruzmaĭkin, A. A. and Sokoloff, D. D. (1990). The Almighty Chance. World Scientific Lecture Notes in Physics 20. World Scientific, River Edge, NJ.