Regularization Effects for Certain Dynamical Systems through Gaussian Noises

It is well known that randomness can be used as an effective tool to turn a priori ill-posed problems into well-posed ones. This is useful both for answering questions at the theoretical as well as the practical levels. Examples of the effectiveness of such an approach are abundant in the fields of optimization, numerical analysis, inverse problems, AI and machine learning, to name a few. On the other hand, continuous-time dynamical systems in the form of Ordinary Differential Equations and the related transport and continuity Partial Differential Equations, the main object of study in this thesis, appear in the modelling of several natural phenomena. A common characteristic of many such models is the lack of regularity of their input-data (i.e. vector fields) which makes the use of classical results, relying for the most part on smoothness assumptions, ineffective. In this work, we study the effect of applying novel regularization maps to certain classical continuous-time dynamical systems with discontinuous vector fields and hence ill-posed a priori. We prove, among other results, that these regularization maps induce a modified dynamics enjoying well-posedness as well as increased stability, with respect to initial conditions. A key aspect of our main result is the preservation of the notion of solution when passing from the original problem to the regularized one