We will first consider interacting particle systems that provide powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. The first model that we will define is a non-linear stochastic PDE that provides a faithful representation of the evolution of the empirical density of a given particle system. This model has a direct applications in the opinion dynamics that will be discussed. Furthermore, we will explain difficulties in numerical approximations of these problems.

Instead of considering many particles, next we will consider just one Brownian particle, but which is now evolving on a random domain. Using the rough path analysis, we will investigate different scaling regimes of this system. As a natural question in this setting is how to present a Gaussian random fields on a sphere. One way to do this is using the so-called spherical harmonics. We will discuss the advantages of this approach and challenges in its generalizations to an arbitrary manifold.