Abstract: Stochastic reaction networks are continuous-time Markov chains typically used in biology, epidemiology, and population dynamics. The goal is to keep track of the abundance of the different reactants over time. What makes them special from a mathematical point of view is the fact that their qualitative dynamics is described by a finite set of allowed transformation rules, referred to as "reaction graph". A long-standing conjecture is that models with a reaction graph composed by a union of strongly connected components are necessarily positive recurrent, meaning that each single state is positive recurrent. In my talk I will discuss why the conjecture makes intuitive sense and why it is difficult to prove it. I will then show how my collaborators and I adapted Forster-Lyapunov techniques to prove the conjecture in two dimensions.