Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A boundedness trichotomy for the stochastic heat equation

Le Chen, Davar Khoshnevisan, and Kunwoo Kim

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We consider the stochastic heat equation with a multiplicative white noise forcing term under standard “intermitency conditions.” The main finding of this paper is that, under mild regularity hypotheses, the a.s.-boundedness of the solution $x\mapsto u(t,x)$ can be characterized generically by the decay rate, at $\pm\infty$, of the initial function $u_{0}$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $\mathbf{\Lambda}:=\lim_{\vert x\vert \to\infty}\vert \log u_{0}(x)\vert /(\log\vert x\vert )^{2/3}$.


Nous nous intéressons à l’équation de la chaleur stochastique avec un bruit blanc multiplicatif sous des <<conditions d’intermittence>> standard. Le résultat principal de cet article est que, sous des hypothèses de régularité raisonnables, le caractère presque sûrement borné de la solution $x\mapsto u(t,x)$ est entièrement déterminé par la vitesse de décroissance en $\pm\infty$ de la condition initiale $u_{0}$. Plus précisément, nous démontrons qu’il existe trois régimes distincts selon la valeur de $\mathbf{\Lambda}:=\lim_{\vert x\vert \to\infty}\vert \log u_{0}(x)\vert/(\log\vert x\vert )^{2/3}$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1991-2004.

Received: 26 October 2015
Revised: 12 July 2016
Accepted: 13 July 2016
First available in Project Euclid: 27 November 2017

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Mathematical Reviews number (MathSciNet)

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]

The stochastic heat equation


Chen, Le; Khoshnevisan, Davar; Kim, Kunwoo. A boundedness trichotomy for the stochastic heat equation. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1991--2004. doi:10.1214/16-AIHP780.

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  • [1] L. Chen and R. C. Dalang. Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions. Stoch. Partial Differ. Equ. Anal. Comput. 2 (3) (2014) 316–352.
  • [2] L. Chen and R. C. Dalang. Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions. Ann. Probab. 43 (6) (2015) 3006–3051.
  • [3] D. Conus, D. Khoshnevisan and M. Joseph. On the chaotic character of the stochastic heat equation, before the onset of intermittency. Ann. Probab. 41 (3B) (2013) 2225–2260.
  • [4] R. C. Dalang. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 (6) (1999) 29 (electronic).
  • [5] K. Dareiotis and M. Gerencsér. On the boundedness of solutions of SPDEs. Stoch. Partial Differ. Equ. Anal. Comput. 3 (1) (2015) 84–102.
  • [6] M. Foondun and D. Khoshnevisan. Intermittence and nonlinear stochastic partial differential equations. Electron. J. Probab. 14 (21) (2009) 548–568.
  • [7] M. Foondun and D. Khoshnevisan. On the global maximum of the solution to a stochastic heat equation with compact-support initial data. Ann. Inst. Henri Poincaré Probab. Stat. 46 (4) (2010) 895–907.
  • [8] M. Joseph, D. Khoshnevisan and C. Mueller. Strong invariance and noise comparison principles for some parabolic SPDEs. Ann. Probab. 45 (1) (2017) 377–403.
  • [9] D. Khoshnevisan. Analysis of Stochastic Partial Differential Equations. Published by the AMS on behalf of CBMS Regional Conference Series in Mathematics 119, (2014) 116. Providence RI.
  • [10] C. Mueller. On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37 (4) (1991) 225–245.
  • [11] C. Mueller. Some tools and results for parabolic stochastic partial differential equations (English summary). In A Minicourse on Stochastic Partial Differential Equations 111–144. Lecture Notes in Math. 1962. Springer, Berlin, 2009.
  • [12] T. Shiga. Ergodic theorems and exponential decay of sample path for certain interacting diffusions. Osaka J. Math. 29 (1992) 789–807.
  • [13] J. B. Walsh. An introduction to stochastic partial differential equations. In Ècole D’èté de Probabilités de Saint-Flour, XIV – 1984 265–439. Lecture Notes in Math. 1180. Springer, Berlin, 1986.