高斯噪声驱动下随机分数阶发展方程温和解的一些逼近结果
Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise
摘要：
研究了一类带高斯噪声的卷积型随机积分方程的空间逼近性质。这样的方程出现，例如，当考虑温和的解决方案的随机分数阶偏微分方程，但也考虑温和的解决方案的经典随机偏微分方程。方程的关键要求是确定性演化算子的光滑性质，这在抛物型问题中是典型的。我们表明，如果一个有机会获得非光滑的数据估计的确定性误差算子及其衍生物的空间离散化过程中，然后获得误差估计的路径Hölder范数的速率，可以读出确定性的差错率。我们通过考虑一类随机分数阶偏微分方程和空间近似的谱Galerkin方法和有限元来说明主要结果。我们还改进了已有的关于随机热方程的结果。
英文摘要：
We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.
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