Second-order fields for stochastic partial differential equations

Abstract

In this article, we study the second-order fluctuation of the solutions of one-dimensional polynomial stochastic partial differential equations (SPDEs) of the form \begin{equation*} (\partial_t - Δ) Φ_{\varepsilon} = -P(Φ_{\varepsilon}) + ξ_{\varepsilon}, \end{equation*} where $P$ is a polynomial of degree greater than or equal to $2$, ${\xi }_{\epsilon }$ is the white-noise after being convoluted (in space) by the heat kernel $K_{\epsilon }=e^{\epsilon \Delta }$. More precisely, taking advantage of the local solutions of pointwise well-posedness of limits $\Phi ={\lim }_{\epsilon \to 0}{\Phi }_{\epsilon }$, we characterise the limit of ${\Phi }^{err}={\lim }_{\epsilon \to 0}{\epsilon }^{-1}({\Phi }_{\epsilon }-\Phi )$ as the solution of a more irregular stochastic partial differential equation. This is performed by applying the Da Prato--Debussche decomposition in the non-linear equation and characterising the limit of each of the terms. We also discuss possible characterisations of second-order fluctuations performed for higher-dimensional SPDEs in the weakly-coupled regime.

11 pages, 1 figure