Singular SPDEs with the Cauchy-Riemann operator on a torus

Abstract

We prove the existence of solution to the following ${\mathbb{C}}^3$-valued singular SPDE on the 2D torus ${\mathbb{T}}^2$: \begin{align} \label{CR} \partial_{\bar z} r = r \times \overline{r} + i \, γ\, {\mathscr W}, \tag{CR} \end{align} where ${\partial }_{\bar{z}}:=\frac{1}{2}({\partial }_x+i{\partial }_y)$ is the Cauchy-Riemann operator on ${\mathbb{T}}^2$, $\mathcal{W}=({\mathcal{W}}_1,{\mathcal{W}}_2,{\mathcal{W}}_3)$ is a real 3D white noise on ${\mathbb{T}}^2$ whose component ${\mathcal{W}}_3$ has zero mean over ${\mathbb{T}}^2$, ${\hat{I}}^3:=({\hat{I}}_1^3,{\hat{I}}_2^3,{\hat{I}}_3^3)$ is an ${\mathbb{R}}^3$-vector and ${\hat{I}}^3\mathcal{W}:=({\hat{I}}_1^3{\mathcal{W}}_1,{\hat{I}}_2^3{\mathcal{W}}_2,{\hat{I}}_3^3{\mathcal{W}}_3)$.