Convergence to equilibrium for stochastic differential equations (SDEs) with multiple invariant measures
Reference Number: 2018-HW-Maths-55
Abstract: The ergodic theory for SDEs is a well developed branch of mathematics. Broadly speaking, such a theory, initiated in the thirties by Birkhoff, deals with the long-time behaviour of SDEs and, more specifically, with determining i) suitable conditions under which the process admits a unique invariant measure -- which, in a probabilistic context, represents the equilibirum of the process -- and ii) studying convergence to such an invariant measure. While a large body of knowledge is available when addressing the study of ergodic processes, the development of a general framework to understand problems with multiple equilibria is at a very early stage. In particular it is well known that ergodic processes will, under appropriate conditions, converge to their unique equilibrium irrespective of the initial configuration, i.e. they will tend to lose memory of the initial datum. This is a desirable property in uncertainty quantification, e.g. in the context of statistical sampling. However such a property is not, in general, satisfied by many non-synthetic systems (birds don't flock just in one direction, the microstructure of nematic crystals aligns to several possible equilibrium configurations) as the initial state does, in general, influence the asymptotic behaviour. This project will deal with the very timely problem of studying stochastic dynamics (mostly diffusion processes) with multiple equilibria. The project will be in collaboration with Imperial College London and the student taking up this research will be encouraged to travel to London for research visits.
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