Strong and Weak Approximation of Semilinear Stochastic Evolution Equations
Strong and Weak Approximation of Semilinear Stochastic Evolution Equations
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DOI：
10.1007/978-3-319-02231-4
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发表时间：
2013-11
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通讯作者：
R. Kruse
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作者：
R. Kruse
这本专著是我博士学位的成果。这是我2008年至2012年在比勒费尔德大学写的论文。它的主要目标是分析的数值方法，近似解随机发展方程（SEEq）的无限维状态空间。因此，我邀请读者加入我在几个迷人的数学领域的交汇点：数值分析，概率论和随机分析，PDE理论和泛函分析。对我来说，研究的数值方法总是有方便的优势，我可以调查我感兴趣的对象不仅在理论上，而且在计算机实验。在怀疑的时刻，数值模拟可能会激励我更努力地寻找理论证明，或者如果我走错了路，它有助于避免在死胡同上浪费时间。计算机实验也是一个持续不断的灵感源泉，它让我在意想不到的现象中跌跌撞撞，对这些现象的理解常常会引发有趣的理论问题。在这方面，SEEqs的数值方法是一个非常有意义的研究对象。即使不改变初始参数，仅仅通过随机强迫项的存在，就可以看到许多不同类型的行为。一个非常有趣的任务是弄清楚哪些现象是由于潜在的随机演化方程，哪些是由于数值方法的应用及其不可避免的错误。为此，当然有必要首先仔细研究随机演化方程及其性质。由于这本书主要是从数值分析的角度写的，我们集中在对数值方案的设计和行为很重要的基本问题上。特别是，我们研究的存在性，唯一性和时空规律的解决方案SEEqs。在熟悉了我们想要近似的对象之后，我们接下来必须考虑近似的目的。我们的数值方案是给出解的一个很好的路径近似，还是仅仅近似地再现某些统计性质？这两种可能性与我们在本专题中研究的两种不同类型的错误有关。一方面，存在所谓的强收敛误差，这是根据随机变量的Lp范数来衡量的。这个误差很小，如果
This monograph grew out of my Ph. D. thesis, which I wrote at Bielefeld University between 2008 and 2012. Its main objective is the analysis of numerical methods, which approximate solutions to stochastic evolution equations (SEEq) on infinite dimensional state spaces. I therefore invite the reader to join me at the meeting point of several fascinating mathematical fields: numerical analysis, probability theory and stochastic analysis, PDE theory, and functional analysis. For me the study of numerical methods always has the convenient advantage that I can investigate my object of interest not only theoretically but also during computer experiments. In moments of doubts, a numerical simulation may motivate me to look harder for a theoretical proof or it helps avoid wasting time on a dead end if I am on the wrong track. Computer experiments also serve as a constant well of inspiration by making me stumbling over unexpected phenomena, whose understanding often raises interesting theoretical questions. To this respect numerical methods for SEEqs are a very grateful study object. Even without changing the initial parameters one may be able to see many different kinds behaviors just by the presence of the random forcing terms. A very interesting task is then to puzzle out which phenomena are due to the underlying stochastic evolution equation and which stems from the application of a numerical method and their inevitable errors.For this it is of course necessary to first have a closer look at stochastic evolution equations and their properties. As this book is mainly written from the viewpoint of a numerical analyst, we concentrate on basic questions which are important for the design and the behavior of numerical schemes. In particular, we study existence, uniqueness, and spatio-temporal regularity of solutions to SEEqs. After having made ourselves familiar with the object which we want to approximate, we next have to think about the aim of the approximation. Shall our numerical scheme give a good pathwise approximation of the solution or shall it just closely reproduce certain statistical properties? The two possibilities are related to the two different kinds of errors which we study in this monograph. On the one hand, there is the so-called strong error of convergence, which is measured in terms of the Lp-norm for random variables. This error is small if