Toronto Probability

This talk is about the large-time behaviour of diffusions on finite-dimensional state spaces. In many applications, one is interested in establishing "ergodicity" or "stability" properties: the process essentially explores all of the state space, time averages of smooth functions of the state tend to space averages with respect to a uniquely identified invariant measure with good properties, and so on. Roughly speaking, such properties are expected when the noise is sufficiently nondegenerate and an appropriate form of compactness holds. In a recent preprint with Y. Bakhtin and L.-S. Young [arXiv:2508.20968], we consider, motivated by applications to scalable networks, diffusions on a certain class of compact, 1- and 2-dimensional domains, assuming that the noise has varying degree of degeneracy on the domain's boundary. Also assuming forms of smoothness and boundary hyperbolicity, we explore all scenarios in which the above ergodicity-type properties fail. Doing so, we are led among other things to consider a notion of what it means for several but not all of the ergodic invariant measures to be dynamically "visible", and to revisit the Foster–Lyapunov technique for proving (positive) recurrence.