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November 2017 A boundedness trichotomy for the stochastic heat equation
Le Chen, Davar Khoshnevisan, Kunwoo Kim
Ann. Inst. H. Poincaré Probab. Statist. 53(4): 1991-2004 (November 2017). DOI: 10.1214/16-AIHP780

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

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

Citation

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Le Chen. Davar Khoshnevisan. Kunwoo Kim. "A boundedness trichotomy for the stochastic heat equation." Ann. Inst. H. Poincaré Probab. Statist. 53 (4) 1991 - 2004, November 2017. https://doi.org/10.1214/16-AIHP780

Information

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

zbMATH: 06847071
MathSciNet: MR3729644
Digital Object Identifier: 10.1214/16-AIHP780

Subjects:
Primary: 60H15
Secondary: 35R60

Rights: Copyright © 2017 Institut Henri Poincaré

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Vol.53 • No. 4 • November 2017
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