Date: 9 June 2025 - 13 June 2025
We will first discuss the general connection, via the voting models, between the branching Brownian motion (BBM) and the Fisher-KPP and other semilinear parabolic equations. This connection, together with the long time asymptotics of the solutions, allows, in particular, to deduce some fine properties of BBM, such as the law of large numbers and fluctuations of the limiting process near the maximum of BBM. In the second half of the lectures, we will discuss some knowledge diffusion models that arise in macroeconomics and lead to mean-field game type of problems that are also related to non-local Fisher-KPP equations.