In this thesis we will study the ergodic measures and the hydrodynamic limit of independent run-and-tumble particle processes, i.e., an interacting particle system for particles with an internal energy source, which makes them move in a preferred direction that changes at random times. We start by providing some basic concepts and theory of Markov processes and interacting particle systems. Afterwards, we define our model on the particle state space $\mathbb{Z}^d \times S$, with $S$ a finite space of internal states, by giving its generator, and we prove a duality result with a similar process which we will use repeatedly throughout this thesis. Then we show that the product Poisson measures with constant parameter are ergodic, and are also the only ergodic probability measures for this process in the space of so-called tempered measures, i.e., measures with bounded factorial moments. Lastly we prove the hydrodynamic limit of this process on $\mathbb{Z} \times S$ by showing that the evolution of the macroscopic density is a weak solution to a PDE.

The normal distribution is a very important distribution in probability theory and statisticsand has a lot of unique properties and characterizations. In this report we look at the proof of two of these characterizations and create counterparts of a normal distribution on abstract spaces, such as vector spaces and groups, which we shall call Gaussians. When we look at R^d, all these Gaussians coincide, along with a Gaussian vector in the normal sense, called multivariate normal. Furthermore, for one Gaussian we prove that it has exponential integrability properties.