in most cases for continuous time stochastic systems which are modeled by SDE, the Lyapunov stability conditions can guarantee the stochastic stability of the system,
another definition In stochastic literature is detailed balance which guarantee the convergence of the probability of the states of SDE to a stationary probability density.
I want to know which one is more strong stability condition?
Dear Mr.Namadchian,
Liapunov stability is dependant on moments which you can indentify in a stationary probability distribution. But the detailed balance gives you fwer equations to calculate the moments. Hence the Liapunov method is theoretically more general.
S.K.Mallick
for S.K.Mallick, S.Raychaudhury, S.Mallick
of the RHMHM School,
USA, Japan, India
Dear Dr. Soumitra K Mallick
I could not understand this sentence:
"Liapunov stability is dependant on moments which you can indentify in a stationary probability distribution "
you mean by Lyapunov stochastic stability we can find the steady state distribution?
I vaguely remembered that the Liapunov function essentially can be represented by its moments which therefore is the Liapunov method and so that should apply to the stochastic version.
S.K.Mallick
With Markov chains having a detailed balance distribution is a stronger condition then having a stationary distribution (that you can check by Lyapunov theory).
Every detailed balance distribution is stationary, but not all stationary distributions are detailed balanced. The proof is very easy. Detail balance means there exist a distribution pi(k) such that
pi(k)q_kj =pi(j)q_jk,
where q_jk is any non-diagonal entry of the generator matrix Q.
Now if on both sides you take the sum over all the k that are different from j you get that stationarity follows (pi Q=0).
You can check it on Durret's book Essentials of stochastic processes, page 130
https://services.math.duke.edu/~rtd/EOSP/EOSP2E.pdf
I'm not sure what happens with diffusions, but it shuold be the same.
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