Singular SPDEs with the Cauchy-Riemann operator on a torus
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| Formato: | Preprint |
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2025
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| author | Brzeźniak, Zdzisław Neklyudov, Mikhail Shamarova, Evelina |
| author_facet | Brzeźniak, Zdzisław Neklyudov, Mikhail Shamarova, Evelina |
| contents | 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 $\partial_{\bar z}: = \frac12(\partial_x + i \partial_y)$ 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 $\mathbb{T}^2$, $γ: = (γ_1,γ_2,γ_3)$ is an $\mathbb{R}^3$-vector and $γ\, {\mathscr W}: = (γ_1 {\scriptstyle {\mathscr W}_1}, γ_2 {\scriptstyle {\mathscr W}_2}, γ_3 {\scriptstyle {\mathscr W}_3})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20075 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular SPDEs with the Cauchy-Riemann operator on a torus Brzeźniak, Zdzisław Neklyudov, Mikhail Shamarova, Evelina Probability Analysis of PDEs 60H15, 35J46 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 $\partial_{\bar z}: = \frac12(\partial_x + i \partial_y)$ 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 $\mathbb{T}^2$, $γ: = (γ_1,γ_2,γ_3)$ is an $\mathbb{R}^3$-vector and $γ\, {\mathscr W}: = (γ_1 {\scriptstyle {\mathscr W}_1}, γ_2 {\scriptstyle {\mathscr W}_2}, γ_3 {\scriptstyle {\mathscr W}_3})$. |
| title | Singular SPDEs with the Cauchy-Riemann operator on a torus |
| topic | Probability Analysis of PDEs 60H15, 35J46 |
| url | https://arxiv.org/abs/2503.20075 |