Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Finite-time singularity of the stochastic harmonic map flow

Antoine Hocquet

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Abstract

We investigate the influence of an infinite dimensional Gaussian noise on the bubbling phenomenon for the stochastic harmonic map flow $u(t,\cdot):\mathbb{D}^{2}\to\mathbb{S}^{2}$, from the two-dimensional unit disc onto the sphere. The diffusion term is assumed to have range one pointwisely in the tangent space $T_{u(t,x)}\mathbb{S}^{2}$, so that the noise preserves the 1-corotational symmetry of solutions. Under the assumption that its space-correlation is of trace class (in some appropriate Hilbert space), we prove that the noise generates blow-up with positive probability. This scenario happens no matter how we choose the initial data, provided it fulfills the latter symmetry assumption.

Résumé

Nous analysons ici l’influence d’un bruit gaussien infini-dimensionnel sur le phénomène de bubbling relatif au flot stochastique des applications harmoniques $u(t,\cdot):\mathbb{D}^{2}\to\mathbb{S}^{2}$, du disque unité vers la sphère. On suppose que le terme de diffusion est ponctuellement de rang un dans le plan tangent $T_{u(t,x)}\mathbb{S}^{2}$, de sorte que le bruit préserve la symétrie 1-corotationnelle des solutions. Sous l’hypothèse que sa corrélation spatiale est de trace finie (dans un espace de Hilbert ah hoc), nous montrons que le bruit engendre une singularité avec probabilité non nulle. Ce scénario se produit indépendamment du choix de la condition initiale, pourvu qu’elle soit 1-corotationnelle.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1011-1041.

Dates
Received: 31 July 2016
Revised: 13 April 2018
Accepted: 13 April 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820840

Digital Object Identifier
doi:10.1214/18-AIHP907

Subjects
Primary: 60H15 (35R60) 58E20: Harmonic maps [See also 53C43], etc. 35K55: Nonlinear parabolic equations 35B44: Blow-up

Keywords
Stochastic partial differential equation Harmonic Maps Blow-up

Citation

Hocquet, Antoine. Finite-time singularity of the stochastic harmonic map flow. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1011--1041. doi:10.1214/18-AIHP907. https://projecteuclid.org/euclid.aihp/1557820840


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