The present proposal is concerned with the study of regularisation-by-noise phenomena. For those born before the 90s, it can be thought of as the classical technique of (not so) gently slapping the TV in order to bring it back to life. A similar phenomenon happens with differential equations. 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. This demonstrates that randomness can have a beneficial impact on dynamical systems.
Equations which need the presence of the noise in order to be well-posed are extremely interesting from mathematical point of view, but 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. Further, there is an increasing interest of the mathematical community in the regularising properties that noise can have in equations arising in fluid dynamics which are not known to be well-posed, such as the 3D Navier-Stokes equation. Up to now, there has been a satisfactorily developed theory for a particular class of noises, so-called Markovian, whose main characteristic is that they do not have memory. However, in practice, most of the systems do have memory. Consequently, non-Markovian noises are day-by-day used in order to model noisy dynamical systems, for example, the so-called rough volatility models in finance. The theory concerning such noises is not so well-developed. The existing techniques strongly rely on the Markovian nature of the underlying problem, hence they cannot be applied to non-Markovian settings. On a more conceptual level, the main challenge lies on a unique feature of this type of noises: they are both "friend and enemy". On the one hand, their roughness provides a regularisation effect, but on the other hand it gives rise to analytically ill-defined mathematical objects.
The aim of the this project is to reconcile these two sides of such noises. We will develop methods that on the one hand will allow us to quantify the regularising properties of the noise and at the same time to handle those analytically ill-defined objects. Further, we will use these properties in order to show well-posedness for a large class of extremely singular equations, both in finite and infinite dimensions, and we will study their numerical approximation. In order to do that, we are planning to introduce a novel method based on very recently obtained Stochastic Sewing techniques combined with tools from Malliavin Calculus and Rough Paths, which will overcome all the limitations of the traditional methods. Other than the results, our main contribution will be the method itself. Since it will not rely on classical tools from Itô's theory, it will be applicable in a variety of settings where these tools are not available, and it will set the foundations for new directions of research.
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