Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations

Le Chen and Kunwoo Kim

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Abstract

In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on $\mathbb{R}$ with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. These results extend Mueller’s comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures with heavier tails than linear exponential growth at ${\pm}\infty$. These results generalize a recent work by Moreno Flores (Ann. Probab. 42 (2014) 1635–1643), who proves the strict positivity of the solution to the stochastic heat equation with the delta initial data. As one application, we establish the full intermittency for the equation. As an intermediate step, we prove the Hölder regularity of the solution starting from measure-valued initial data, which generalizes, in some sense, a recent work by Chen and Dalang (Stoch. Partial Differ. Equ. Anal. Comput. 2 (2014) 316–352).

Résumé

Dans ce papier, nous montrons un principe de comparaison trajectoriel pour l’équation de la chaleur stochastique, fractionnaire, nonlinéaire sur $\mathbb{R}$ avec une donnée initiale à valeur mesure. Nous donnons des estimations quantitatives de la proximité à zéro d’une solution. Ces résultats étendent le principe de comparaison de Mueller pour l’équation de la chaleur stochastique et permettent de considérer des données initiales plus générales telles que des mesures de Dirac et des mesures à queue plus lourde qu’une croissance exponentielle linéaire en ${\pm}\infty$. Ces résultats généralisent un travail récent par Moreno Flores (Ann. Probab. 42 (2014) 1635–1643), qui a prouvé la stricte positivité de l’équation de la chaleur stochastique partant d’un Dirac. Comme application, nous établissons la complète intermittence pour l’équation. Dans une étape intermédiaire, nous prouvons la régularité Hölder de solutions partant d’une donnée initiale à valeur mesure ce qui généralise, dans un certain sens, un travail récent de Chen and Dalang (Stoch. Partial Differ. Equ. Anal. Comput. 2 (2014) 316–352).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 358-388.

Dates
Received: 3 October 2014
Revised: 4 October 2015
Accepted: 5 October 2015
First available in Project Euclid: 8 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1486544895

Digital Object Identifier
doi:10.1214/15-AIHP719

Mathematical Reviews number (MathSciNet)
MR3606745

Zentralblatt MATH identifier
1361.60049

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60G60: Random fields 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Keywords
Nonlinear stochastic fractional heat equation Parabolic Anderson model Comparison principle Measure-valued initial data Stable processes

Citation

Chen, Le; Kim, Kunwoo. On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 358--388. doi:10.1214/15-AIHP719. https://projecteuclid.org/euclid.aihp/1486544895


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