A one-dimensional non-local singular SPDE

Abstract

We examine in this article the one-dimensional, non-local, singular SPDE \begin{equation*} \partial_t u \;=\; -\, (-\Delta)^{1/2} u \,-\, \sinh(\gamma u) \,+\, \xi\;, \end{equation*} where $\gamma\in \mathbb{R}$, $(-\Delta)^{1/2}$ is the fractional Laplacian of order $1/2$, $\xi$ the space-time white noise in $\mathbb{R} \times \mathbb{T}$, and $\mathbb{T}$ the one-dimensional torus. We show that for $0<\gamma^2<\pi/7$ the Da Prato--Debussche method applies. One of the main difficulties lies in the derivation of a Schauder estimate for the semi-group associated to the fractional Laplacian due to the lack of smoothness resulting from the long range interaction.