AI 中文总结
研究随机艾伦 - 卡恩方程的统计推断，通过分析其温和解的空间二次变差渐近行为确定极限值，利用解的分解及热核分析得出\(Y\)的正则性，基于此开发参数估计程序。
AI 中文摘要
本文探讨随机偏微分方程的统计推断。研究了由时空白噪声驱动的随机艾伦 - 卡恩方程并分析其温和解。主要关注解的空间二次变差的渐近行为，确定了其精确极限值。作为应用，基于这些渐近结果开发了参数估计程序。证明了唯一解可分解为\(u = X + Y\)，其中\(X\)是线性随机热方程的解，\(Y\)表示非线性效应。通过对热核及其缩放行为的详细分析，得出\(Y\)在空间和时间变量上的赫尔德连续性性质，表明\(Y\)比\(X\)具有更高的正则性。这种分解和正则性估计是基于解的二次变差渐近行为开发参数估计程序的关键要素。
英文摘要
This paper addresses statistical inference for stochastic partial differential equations. We study the stochastic Allen-Cahn equation driven by space-time white noise and analyze its mild solution. Our main focus is the asymptotic behavior of the spatial quadratic variation of the solution, for which we establish the exact limiting value. As an application, we develop parameter estimation procedures based on these asymptotic results. We prove that the unique solution can be decomposed as u = X + Y where X denotes the solution of the linear stochastic heat equation and Y accounts for the nonlinear effects. Exploiting a detailed analysis of the heat kernel and its scaling behavior, we derive Hölder continuity properties of Y in both spatial and temporal variables, showing that Y exhibits substantially higher regularity than X. This decomposition and the resulting regularity estimates are key ingredients in the development of parameter estimation procedures based on the asymptotic behavior of quadratic variations of the solution.
