A few problems on stochastic geometric wave equations

In this thesis, we study three problems on stochastic geometric wave equations. First, we prove the existence of a unique local maximal solution to an energy critical stochastic wave equation with multiplicative noise on a smooth bounded domain $\mathcal{D} \subset \mathbb{R}^2$ with exponential nonlinearity. The main ingredients in the proof are appropriate deterministic and stochastic Strichartz inequalities which are derived in suitable spaces.

In the second part, we verify a large deviation principle for the small noise asymptotic of strong solutions to stochastic geometric wave equations. The method of proof relies on applying the weak convergence approach of Budhiraja and Dupuis to SPDEs where solutions are local Sobolev spaces valued stochastic processes.

The final result contained in this thesis concerns the local well-posedness theory for geometric wave equations, perturbed by a fractional Gaussian noise, on one dimensional Minkowski space $\mathbb{R}^{1+1}$ when the target manifold $M$ is a compact Riemannian manifold and the initial data is rough. Here, to achieve the existence and the uniqueness of a local solution we extend the theory of pathwise stochastic integrals in Besov spaces to two dimensional case.