论文标题
来自本地测量的半连接SPDE的参数估计
Parameter estimation for semilinear SPDEs from local measurements
论文作者
论文摘要
这项工作有助于有限的文献来估计由加性噪声驱动的非线性SPDE的扩散率或漂移系数。假设该溶液是在空间和有限的时间间隔内局部测量的，我们表明，在Altmeyer中引入的增强最大似然估计量，Reiss（2020）（2020）保留了其渐近性特性，用于满足某些抽象和可验证的条件的半连续性SPD。渐近结果的证明是基于在线性和非线性零件中拆分解决方案的，以及$ l^p $ - 空格中的精细规律性属性。获得的一般结果应用于特定的方程类别，包括随机反应扩散方程。随机汉堡方程作为一阶非线性的一个例子，是一般结果的一个有趣的边界案例，并由维纳尔混乱的扩展对待。我们以验证理论结果的数值示例结束。
This work contributes to the limited literature on estimating the diffusivity or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that the solution is measured locally in space and over a finite time interval, we show that the augmented maximum likelihood estimator introduced in Altmeyer, Reiss (2020) retains its asymptotic properties when used for semilinear SPDEs that satisfy some abstract, and verifiable, conditions. The proofs of asymptotic results are based on splitting the solution in linear and nonlinear parts and fine regularity properties in $L^p$-spaces. The obtained general results are applied to particular classes of equations, including stochastic reaction-diffusion equations. The stochastic Burgers equation, as an example with first order nonlinearity, is an interesting borderline case of the general results, and is treated by a Wiener chaos expansion. We conclude with numerical examples that validate the theoretical results.
