Low regularity solutions to the stochastic geometric wave equation driven by a fractional Brownian sheet

Zdzisław Brzeźniak,Nimit Rana

doi:10.5802/crmath.38

Abstract

We announce a result on the existence of a unique local solution to a stochastic geometric wave equation on the one dimensional Minkowski space ℝ 1+1 with values in an arbitrary compact Riemannian manifold. We consider a rough initial data in the sense that its regularity is lower than the energy critical.

Highlights

The existence and the uniqueness of a global solution, in the strong and weak sense, for the stochastic geometric wave equations (SGWEs) on the Minkowski space R1+m, m ≥ 1, with the target manifold (N, g ) being a suitable n-dimensional Riemannian manifold, e.g. a sphere, has been established under various sets of assumptions by the first named author and M
To the best of our knowledge, the most general result in the case m = 1, is a construction of a global Hl1oc (N ) × L2loc (T N )-valued weakly continuous solution of SGWE, where T N denotes the tangent bundle of N, see [2]
The purpose of this note is to present a method by which we can prove the existence of a unique local solution to SGWE with m = 1 in the case of the initial data belonging to Hlsoc (N ) ×

Summary

Introduction

The existence and the uniqueness of a global solution, in the strong and weak sense, for the stochastic geometric wave equations (SGWEs) on the Minkowski space R1+m, m ≥ 1, with the target manifold (N , g ) being a suitable n-dimensional Riemannian manifold, e.g. a sphere, has been established under various sets of assumptions by the first named author and M. The purpose of this note is to present a method by which we can prove the existence of a unique local solution to SGWE with m = 1 in the case of the initial data belonging to Hlsoc (N ) ×. In particular, we generalize the corresponding deterministic theory result of [8] to the stochastic setting, as well as the results of [1,2,3] to the wave maps equation with low regularity initial data (i.e. s < 1) and fractional (both in time and space) Gaussian noise. ISSN (electronic) : 1778-3569 https://comptes- rendus.academie- sciences.fr/mathematique/

Problem formulation

Relevant notation and function spaces

Hαs Hβδ

The main result: the local well-posedness