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 \, \gamma \, {\mathscr W}, \tag{CR} \end{align} where $\partial_{\bar z}: = \frac12(\partial_x + i \partial_y)$ is the Cauchy-Riemann operator on $\mathbb{T}^2$, ${\mathscr W} = ({\scriptstyle {\mathscr W}_1}, {\scriptstyle {\mathscr W}_2}, {\scriptstyle {\mathscr W}_3})$ is a real 3D white noise on $\mathbb{T}^2$ whose component ${\scriptstyle {\mathscr W}_3}$ has zero mean over $\mathbb{T}^2$, $\gamma: = (\gamma_1,\gamma_2,\gamma_3)$ is an $\mathbb{R}^3$-vector and $\gamma \, {\mathscr W}: = (\gamma_1 {\scriptstyle {\mathscr W}_1}, \gamma_2 {\scriptstyle {\mathscr W}_2}, \gamma_3 {\scriptstyle {\mathscr W}_3})$.