The connections between probability theory on the one hand and differential and integral equations on the other, are so numerous and diverse that the task of presenting them in a comprehensive and connected manner appears almost impossible.Mark Kac, 1951
The interplay between differential equations and probability has only become richer and more profound since Mark Kac wrote the above lines, close to 70 years ago. The thematic program focuses on two aspects of this interplay.
The first focus is on how the introduction of randomness into PDEs (e.g., through random initial conditions, random environments, or random forcing) affects their long-term behaviour. This provides for more flexible and realistic physical models which can explain a wider range of observed behaviour. Moreover, in some cases, randomness is an unavoidable feature of any reasonable PDE model for the underlying system. This is the situation in stochastic control problems or PDE approximations for optimization problems involving multiple agents with incomplete or imperfect information.
The second aspect is how both subjects are used to build mathematical models of group dynamics, and on the interplay between such models. The phrase “group dynamics” could refer, for example, to species migration, the spread of a virus, or the propagation of electrons through an inhomogeneous medium, to name a few examples. Very commonly, the stochastic processes track the corresponding PDEs in the large-population limit. When this can be proved to hold, it allows for information to be passed between the two mathematical subjects.