Imagine tossing a coin in a game of heads or tails many times. We expect that the coin will come up heads roughly 50% of the time. However, because the outcome of each toss is random, there will be some fluctuation around this 50%. One of the main results of probability theory is that this fluctuation follows the normal distribution. The same logic applies to other random experiments, such as rolling a die. This suggests that the normal distribution is a universal object independent of the specific details of the random experiment.

In this project, we will investigate the universality of more complex random processes often used to model phenomena in physics. We will focus on three different types of systems: processes with geometric constraints that induce global correlations in time, processes interacting with a random environment, and singular stochastic differential equations.