A non-linear wave equation with fractional perturbation

We study a $d$-dimensional wave equation model ($2\le d\le 4$) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter $H=(H_0,\dots ,H_d)\in (0,1{)}^{d+1}$ of the noise: if $\sum_{i=0}^d H_i > d-\frac12$,thentheequationcanbetreateddirectly,whileinthecase$d-\frac34<\sum_{i=0}^d H_i\leq d-\frac12$, the model must be interpreted in the Wick sense, through a renormalization procedure. Our arguments essentially rely on a fractional extension of the considerations of \cite{gubinelli-koch-oh} for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.