Fahim K, Hausenblas E, Kovács M
Department of Mathematics, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya, 60111 Indonesia.
Department of Mathematics, Montanuniversity Leoben, 8700 Leoben, Austria.
Stoch Partial Differ Equ. 2023;11(3):1044-1088. doi: 10.1007/s40072-022-00250-0. Epub 2022 Apr 26.
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.
我们研究了一类具有高斯噪声的卷积型随机积分方程的空间逼近质量。例如，当考虑随机分数阶偏微分方程的温和解时，以及当考虑经典随机偏微分方程的温和解时，都会出现此类方程。这些方程的关键要求是确定性演化算子的平滑性质，这在抛物型问题中是典型的。我们表明，如果能够获得确定性误差算子及其空间离散化过程导数的非光滑数据估计，那么就可以在路径 Hölder 范数中获得误差估计，其速率可以从确定性误差速率中读出。我们通过考虑一类随机分数阶偏微分方程以及通过谱伽辽金方法和有限元进行的空间逼近来说明主要结果。我们还改进了关于随机热方程的现有结果。
