Can we make long-term predictions?

该项目将包括证明随机微分方程解的近似在时间误差范围内是一致的，从通过多尺度方法获得的结果开始。为了做到这一点，常用的工具在马尔可夫过程，马尔可夫半群理论，随机分析，数值方法和概率论将被利用。当建模生物过程和多尺度方法时，多尺度方法特别相关。了解我们可以从我们的近似值中获得的误差界限将允许更好地量化不确定性，以及更严格的预测。边界在时间上是一致的，这将使我们能够理解什么时候我们可以期望我们的近似一直保持良好，什么时候它们会随着时间的推移而恶化。当前的错误边界通常适用于有限时间窗口。也就是说，我们已经证明了许多过程和模型的近似值在t时间之前是一个很好的近似值。该项目的新颖之处在于获得与时间无关的边界，以允许长期近似值和更大的预测确定性。这个项目的首要目标是产生一种新的理论，包括实用的标准，以理解何时可以通过数值方案或其他程序近似给定的随机动力学，并且误差不会随时间增加。这个项目包含两个“子项目”。一个考虑通过数值方案产生的近似，另一个考虑通过其他程序产生的近似；在后一种情况下，它将集中于平均或均匀化过程。该项目不会特别关注应用，但这些问题受到数学生物学应用的启发，特别是蜂群，在工程、材料科学、物理等领域有许多应用。

The project will consist of proving uniform in time error bounds on approximations to the solutions of Stochastic differential equations, beginning with those obtained through multiscale methods. In order to do this, common tools in Markov processes, Markov semigroup theory, stochastic analysis, numerical methods, and probability will be utilised. Multiscale methods are particularly relevant when modelling biological processes and multiscale methods. Understanding the error bounds that we can obtain on our approximations will allow better quantification of uncertainty, as well as tighter predictions. The bounds being uniform in time will allow an understanding of when we can expect our approximations to hold well for all time, and when they will deteriorate over time.Current error bounds often hold for finite-time windows. That is, we have proven for many processes and models that the approximation is a good one up until some time T. The novel nature of the project is in obtaining bounds that are independent of time, to allow for long term approximations and greater certainty of predictions.The overarching goal of this project is to produce a novel theory, including practical criteria, to understand when a given random dynamics can be approximated - either via numerical schemes or via other procedures - with an error which does not increase in time. This project contains two "sub-projects". One considering approximations produced via numerical schemes and one considering approximations produced via other procedures; in the latter case it will in concentrate on averaging or homogenization procedures.The project will not be focussing on applications specifically but these problems are inspired by applications to mathematical biology, swarming in particular, with a number of applications in engineering, material science, physics etc.