Imperial College London
Applied Mathematics

Large Deviations: A mathematical theory with important practical applications.

The primary aim of the project is to study phenomena in rare events (large deviation theory, LDT) in complex models of physical phenomena, described by many-particle systems or stochastic partial differential equations. After determining the typical behaviour of a model , further insight is gained by quantifying the probabilities of atypical events are. Informally, LDT expresses that "unlikely events happen in the least unlikely way possible": the quantification of ‘least unlikely’ is given by finding a suitable rate function describing the exponential decay rate of probabilities. Outside of its mathematical interest, LDT is the fundamental Ansatz of Macroscopic Fluctuation Theory in the statistical physics of non-equilibrium systems, and LDT has been applied in option pricing algorithms, engineering and climate science. In mechanical engineering, LDT has been used to study the most probable ways in which mechanical failure can occur, or to recognise early- warning signs, and in climatology, the LDT is used in describing high impact events and to compute extreme values in heat waves.

A common theme in all of these applications is that the exponential decay rate of probabilities must be the same, for all possible trajectories, in the upper and lower bounds. In contrast, the precise theorems for canonical models often prove an estimates with differing decay rates, which only agree on sufficiently 'well-behaved' trajectories. This restriction could lead to wrongly accounting for exactly the interesting behaviour (mechanical failure, catastrophic climate events or stock market shocks…) one most needs to understand in applications. The focus of the project will be to investigate the problem of finding matching bounds, or exhibiting violations of the naturally predicted rate function, in a number of concrete, physically motivated models, which may be useful toy cases for more complex and realistic models.