发表机构
IMPA; USP Instituto de Matemática, Estatística e Ciência da Computação（巴西国家纯粹与应用数学研究所; 圣保罗大学数学、统计与计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Da Prato-Debussche分解刻画一维多项式随机偏微分方程解的二次涨落极限，并讨论高维弱耦合情形下的推广。
AI 中文摘要
本文研究了一维多项式随机偏微分方程（SPDEs）解的二次涨落，其形式为 \\((\\(\partial_t\\) - \\(\Delta\\)) \\(\Phi_{\varepsilon}\\) = -P(\\(\Phi_{\varepsilon}\\)) + \\(\xi_{\varepsilon}\\)\\)，其中 \\(P\\) 是次数大于或等于 2 的多项式，\\(\xi_{\varepsilon}\\) 是白噪声与热核 \\(K_{\varepsilon}\\) = \\(e^{\varepsilon \Delta}\\) 在空间上卷积后的结果。更确切地说，利用极限 \\(\Phi\\) = \\(\lim_{\varepsilon \to 0}\\)\\(\Phi_{\varepsilon}\\) 的逐点适定性的局部解，我们将 \\(\Phi^{err}\\) = \\(\lim_{\varepsilon \to 0}\\) \\(\varepsilon^{-1}\\)(\\(\Phi_{\varepsilon}\\) - \\(\Phi\\)) 的极限刻画为一个更不规则的随机偏微分方程的解。这是通过将 Da Prato--Debussche 分解应用于非线性方程并刻画每一项的极限来实现的。我们还讨论了在弱耦合机制下高维 SPDEs 的二次涨动的可能刻画。
英文摘要
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 $2$, $ξ_\varepsilon$ is the white-noise after being convoluted (in space) by the heat kernel $K_\varepsilon = e^{\varepsilon Δ}$. More precisely, taking advantage of the local solutions of pointwise well-posedness of limits $Φ= \lim_{\varepsilon \to 0}Φ_{\varepsilon}$, we characterise the limit of $Φ^{err}=\lim_{\varepsilon \to 0} \varepsilon^{-1}(Φ_{\varepsilon}-Φ)$ 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.
Comments11 pages, 1 figure
