Arsenic&Lace said:are there methods to treat this as a "coarse grained" Markov chain for which one can then compute transition rates between the classes?
A Markov chain is a mathematical model used to describe a sequence of events in which the probability of each event depends only on the state of the previous event. It is a type of stochastic process that follows the Markov property, which states that the future is independent of the past given the present state. In simpler terms, a Markov chain is a system in which the future is determined solely by the current state.
In a Markov chain, classes are defined as groups of states that can be reached from one another. These states are considered to be "communicating" with each other because there is a non-zero probability of transitioning from one state to another. The classes in a Markov chain are important because they help determine the long-term behavior of the system.
The communication between classes in a Markov chain is determined by the existence of a non-zero probability of transitioning from one class to another. This means that there must be a path of states connecting the two classes, and each of these states must have a non-zero probability of transitioning to the next state in the path. If this condition is met, then the two classes are said to be communicating.
The communicating classes in a Markov chain play a crucial role in determining the long-term behavior of the system. If all states in a Markov chain belong to a single communicating class, then the chain is said to be irreducible, and there is a unique stationary distribution. However, if the chain has multiple communicating classes, then the long-term behavior of the system will depend on the initial state and the transition probabilities between the classes.
If a Markov chain has multiple communicating classes, then the existence of a unique stationary distribution depends on the properties of the transition probabilities between the classes. If the transition probabilities between classes are non-zero, then the chain is said to be positive recurrent, and a unique stationary distribution exists. However, if the transition probabilities between classes are all zero, then the chain is said to be null recurrent, and a unique stationary distribution does not exist.