[PDE & Applications seminar] Ivan Kryven: Partial differential equations leading to random graph models

15 juni 2023 16:00 t/m 17:00 - Locatie: Snijderszaal LB 01.010 EEMCS | Zet in mijn agenda

In their limiting regime, many random models can be reduced to an effective macroscopic differential equation. A famous example of such connection is the Brownian motion, which can be represented by a Laplace operator in the limit. This example is a special case of the Fayman-Kac formula that connects random walks and linear partial differential equations (PDEs). It turns out that only a few analogies of this sort are known for non-linear PDEs. At the same time non-linear PDEs may feature peculiar behaviour, for example, their solution may fail to exist outside of a finite interval of time and characterising this time is typically a challenge.

In this talk we will follow the reverse intuition: we will start with an example of a very specific non-linear particular differential equation and show that the random model that converges to it is the well-known Erdös-Rényi random graph. We will then show that by naturally relaxing the requirements on our PDE, we can arrive at a larger class of random graph models. Interestingly, this analogy can be further exploited to derive existence domains for the non-linear PDEs without explicitly knowing their analytical solution (in the classical sense).

Joint work with Jochem Hoogendijk.