The talk will be devoted to a (partial) presentation of the results of \cite{deya-wave,deya-wave-2} about the following non-linear $2D$ stochastic wave model:

$$\begin{equation}\label{2d-quadratic-wave}\left\{\begin{array}{l}\partial^2_t u-\Delta u= u^2+\dot{B} \quad , \quad t\in [0,T] \ , \ x\in \mathbb{R}^2 \ ,\\u(0,.)=\phi_0 \quad , \quad \partial_t u(0,.)=\phi_1 \ ,\end{array}\right.\end{equation}$$

where $\phi_0,\phi_1$ are (deterministic) initial conditions in an appropriate Sobolev space and $\dot{B}:=\partial_t\partial_{x_1}\partial_{x_{2}} B$, for some space-time fractional Brownian motion $B$ of Hurst index $(H_0,H_1,H_2) \in (0,1)^{3}$. \smallskip

The model has been treated by Gubinelli, Koch and Oh (see \cite{gubinelli-koch-oh}) in the specific white-noise situation, that is when $H_0=H_1=H_2=\frac12$. Our objective in \cite{deya-wave,deya-wave-2} was to generalize these considerations to every $(H_0,H_1,H_2)\in (0,1)^3$ such that $H_0+H_1+H_2>1$ (which turns out to be an optimal condition). \smallskip

Such an extension allows us (in particular) to offer a better perspective on the change-of-regime phenomenon behind: if $H_0+H_1+H_2>\frac32$, then the equation can be directly solved in the classical (mild) sense, while if $H_0+H_1+H_2\leq \frac32$, then the model can only be handled through a renormalization procedure. \smallskip

Besides, our strategy to treat somehow relies on a similar {\it splitting} as in rough paths theory: we first emphasize the role of an explicit

stochastically-defined object $\mathbf{\Psi}=\mathbf{\Psi}(\dot{B})$ at the core of the dynamics (the analog of a rough path, to some extent), then reformulate the equation in terms of $\mathbf{\Psi}$ and settle a deterministic fixed-point argument to solve it.

A. Deya: A non-linear wave equation with fractional perturbation. {\it Ann. Probab.} {\bf 47} (2019), no. 3, 1775-1810.

A. Deya: On a non-linear 2D fractional wave equation. To appear in {\it Ann. Inst. H. Poincar{\'e} Probab. Statist}.

M. Gubinelli, H. Koch and T. Oh: Renormalization of the two-dimensional stochastic non linear wave equations. {\it Trans.Amer. Math. Soc.} {\bf 370} (2018), 7335-7359.