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.

The project set out to advance the state of the art understanding of the "regularisation by noise" in stochastic differential equations. This mathematical term refers to the phenomenon that a dynamic with a very irregular tendency (also referred to as a singular drift) can be assisted by an external noisy force, whose random effects assist in avoiding the irregular traps of the motion. Our objective was to develop a precise theoretical framework that describes which types of randomness provides what amount of smoothing effect. The theoretical foundations also underpin the numerical analysis of computational methods that are used to simulate such stochastic systems. We have achieved major progress in sharp characterisation of regularisation by fractional Brownian motions. These random processes represent anomalous diffusions with long-range memory and arise in many applications including turbulence, hydrology, anomalous polymer dynamics, diffusion in living cells, and rough volatility models in finance. Results of the project give very precise descriptions of the regularisation effects by these processes: among others, we provided sharp criteria on the admissible singularities in the dynamics, path-by-path (i.e. for every single realisation of the randomness) well-posedness, and an efficient statistical description of the solutions. We have also obtained significant results in the analysis of numerical schemes. Here our goal was to understand how the commonly used computational algorithms perform, when used to simulate solutions of equations with singular drift. By deriving rigorous estimates on the error made by the algorithm, we can guarantee their efficiency. This has been achieved in several interesting and challenging settings, including processes with random jumps (as opposed to the continuous motion considered previously) or concerning systems whose evolution has not only a time dimension, but a space dimension as well. Overall, both on the theoretical and the computational side we have developed novel "stochastic sewing" methods, leading to numerous new results, and laying the foundation for future research in the field.