Can we make long-term predictions?

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.