  In this paper, we present criteria for non-exponential ergodicity of
continuous-time Markov chains on a countable state space. These criteria can be
verified by examining the ratio of transition rates over certain paths. We
applied this path method to explore the non-exponential convergence of
microscopic biochemical interacting systems. Using reaction network
descriptions, we identified special architectures of biochemical systems for
non-exponential ergodicity. In essence, we found that reactions forming a cycle
in the reaction network can induce non-exponential ergodicity when they
significantly dominate other reactions across infinitely many regions of the
state space. Interestingly, the special architectures allowed us to construct
many detailed balanced and complex balanced biochemical systems that are
non-exponentially ergodic. Some of these models are low-dimensional bimolecular
systems with few reactions. Thus this work suggests the possibility of
discovering or synthesizing stochastic systems arising in biochemistry that
possess either detailed balancing or complex balancing and slowly converge to
their stationary distribution.
