Non-local equations are currently much studied in modern analysis, in a cross-fertilising interaction with other disciplines such as fractional calculus, stochastic processes, and functional analysis. While PDE feature differential operators, non-local equations are integro-differential equations related to pseudo-differential operators. Differential operators have the property that the evaluation of a function on which they act depends on each point individually, while this dependence extends to all other points in space for non-local operators. This fundamental difference generates a whole range of new qualitative behaviours, unseen in the case of PDE, and also poses new challenges in the mathematical description of such operators and equations.
In this project our aim is to study various analytic properties of solutions of elliptic or parabolic non-local equations. Since often the related operators are generators of Markov processes, a probabilistic representation opens the alternative of a stochastic processes-based analysis beside using direct analytic tools. The project is open to all these possible approaches, and will also connect with the topics of the forthcoming thematic programme "Fractional Differential Equations" proposed and co-organised by the project supervisor, and to be hosted by the Isaac Newton Institute, Cambridge, in 2021.
- Applicants should have, or expect to achieve, at least a 2:1 Honours degree (or equivalent) in mathematics, with a firm background and, desirably, extended (e.g., degree project level) experience with general analysis.
- A relevant Master's degree and / or experience in one or more of the following will be an advantage: Functional Analysis, Stochastic Analysis, Stochastic Processes.
All applications should be made online. Under programme name select School of Science. Please quote reference number: MA/JL-Un2/2019
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