Kelly's lemma
In probability theory, Kelly's lemma states that for a stationary continuous-time [Markov chain], a process defined as the time-reversed process has the same stationary distribution as the forward-time process. The theorem is named after Frank Kelly.
Statement
For a continuous time Markov chain with an at most countable state space and transition-rate matrix, if we can find a set of non-negative numbers and a positive measure that satisfy the following conditions:then are the rates for the reversed process and is proportional to the stationary distribution for both processes.
Proof
Given the assumptions made on the and we haveso the global balance equations are satisfied and the measure is proportional to the stationary distribution of the original process.
By symmetry, the same argument shows that is also proportional to the stationary distribution of the reversed process.