Regularisation-by-noise in finite and infinite dimensions

Regularisation-by-noise is a branch of stochastic analysis that studies the following phenomenon: Certain dynamical systems tend to behave better when a source of randomness is present. Let us give an example. One of the main concerns in the study of differential equations is the so-called well-posedness, that is, the existence and the uniqueness of solutions. However, there are many equations that suffer from lack of well-posedness, that is, they might have multiple (in fact infinitely many) solutions or might not have a solution at all. A remarkable result in mathematics states that for a large class of those equations, well-posedness can be retrieved provided that the system is perturbed by a random (stochastic), sufficiently rough force.

Equations of this type, which need the presence of the noise in order to be well-posed, are very interesting from a mathematical point of view. In addition, their importance goes beyond mathematics as they are increasingly used in the applied sciences. Among others, they are used in engineering in order to simulate transport-diffusion phenomena, in finance for modelling equity markets, and in neuroscience for modelling interacting neurons. Hence, the study of well-posedeness, qualitative properties of their solutions, and their numerical approximations is an important challenge.

The aims of this project are the following:
1. Develop techniques which will allow to quantify the regularising properties of rough noises.
2. Use these techniques in order to study the well-posedness of ordinary and partial differential equations that exhibit regularisation by noise phenomena.
3. Study the numerical approximation of solutions of such equations.