## Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

### A boundedness trichotomy for the stochastic heat equation

#### Abstract

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 $\mathbf{\Lambda}:=\lim_{\vert x\vert \to\infty}\vert \log u_{0}(x)\vert /(\log\vert x\vert )^{2/3}$.

#### Résumé

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

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

Dates
Revised: 12 July 2016
Accepted: 13 July 2016
First available in Project Euclid: 27 November 2017

https://projecteuclid.org/euclid.aihp/1511773735

Digital Object Identifier
doi:10.1214/16-AIHP780

Mathematical Reviews number (MathSciNet)
MR3729644

Zentralblatt MATH identifier
06847071

Keywords
The stochastic heat equation

#### Citation

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. https://projecteuclid.org/euclid.aihp/1511773735

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