Metastability is a phenomenon observed in stochastic systems which stay in a false-equilibrium within a region of its state space until the occurrence of a sequence of rare events that leads to an abrupt transition to a different region. This behaviour is characteristic of some interacting particle systems in statistical physics, although there are models in other contexts, such as perturbations of dynamical systems, that present this behaviour. Recently, a necessary and sufficient condition for the metastability of Markov processes in terms of the resolvent of the process has been proved based on the martingale representation of the process. This new technique to establish metastability has the potential for application in many models in areas such as physics and economics.

Projects will involve understanding the resolvent approach to metastability and applying it to Markov processes, in particular interacting particle systems. A background in probability theory and stochastic processes is necessary.

References

Landim, C.; Marcondes, D.; Seo, I. A resolvent approach to metastability. Journal of the European Mathematical Society, pp.1-56. 2023

Landim, C.; Marcondes, D.; Seo, I. Metastable behavior of weakly mixing Markov chains: The case of reversible, critical zero-range processes. Annals of Probability, 51 (1) 157 - 227. 2023