Regularisation by noise in discrete and continuous systems

When a system is stuck in an unfavourable state, an external force can push it out of it and the system goes on to behave in a much more favourable way. One may think of the classical example of an old TV screen seemingly stuck on a grainy picture, which is suddenly fixed if one "perturbs" the TV device with an appropriately placed hit. In the mathematical context of the project, the role of these perturbations is played by random processes. A wide variety of real-world phenomena is described by random processes. These dynamics are characterised by being driven by a very large number of very small random influences. For example, in financial markets the cumulative effects high-frequency micro-transactions influence the evolution of the prices of financial instruments. Leveraging how these small random oscillations force stochastic processes into favourable behaviour is called regularisation by noise. Our project studies the theory, analysis, and computational treatment of stochastic differential equations with such effects. We set out to advance and employ state of the art mathematical tools to develop the theoretical foundations of how various stochastic processes provide regularisation. Furthermore, we exploit the regularising effects in numerical methods in order to achieve better computational efficiency in the simulation of the random models.