# Stochastic stability and convergence

Consider a Markov process $X$ on $\mathbb R$. Suppose that $X^2$ is $\mathsf P_x$-supermartingale for any $x\in \mathbb R$. If we want that for some neighborhood $U_0$ of $x=0$ holds: for each $x\in U_0$ a condition $X_0 = x$ implies $$\lim\limits_{n\to\infty}X_n= 0$$ then there is a trivial counterexample provided by a process $X_0 = X_1=\dots=X_n=\dots$

Are there more strong conditions on the $X^2$ rather than the supermartingale property that imply local asymptotic stability of an origin?

Some clarification:

1. What I am exactly interested in, are the properties of $X^2$ or $|X|$ described in the terms of the transition semigroup of the process $X$.

2. I am interested in all types of convergence $X_n\to 0$.

3. I wonder if there are results exactly for the discrete-time setting, but I would be happy also if you could refer me to the ones in the continuous time.

4. The book I have at my hand is Kushner, "Stochastic Stability and Control" (1967) which does not fully cover these questions, also I expect that there are more recent results in this field.

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I changed the start of your second sentence. It sounded like you were claiming that the square of any Markov process is a supermartingale. – Byron Schmuland Aug 15 '11 at 15:41
@Byron: oh, that's right ) thank you. – Ilya Aug 15 '11 at 16:21
@Gortaur : Hi, I am not sure I fully get the question. So let me sum up: 1/You have a Markov process $X_t^x$ such that $(X_t^x)^2$ is supermartingale for all x. 2/ You are looking for some "conditions" for the limit of $X_t^x$ as $t\to \infty$ goes to $0$ (in any sense). 3/The conditions must hold over some set $U_0$ which includes the point $0$. 4/Those conditions should hold for the processes $X^2$ or $|X|$ using the semigroup of $X$. 5/ Discrete time setting would be best. Is that what you are looking for or do I misunderstand something ? – TheBridge Aug 16 '11 at 11:53
@TheBridge: you way of asking gives a guess that you know the answer ) Yes, that's what I'm looking for. – Ilya Aug 16 '11 at 12:16
@Gortaur : Unfortunately I don't, but the question is interesting. – TheBridge Aug 17 '11 at 11:28

I actually wrote a paper some time ago, looking into this question. There the setting is a bit more general: instead of $\Bbb R$ there I work with a general Polish space (local compactness is required for stronger results). There are some equivalence results for convergence, and workable sufficient conditions are provided by means of Lyapunov-like functions, see Lemma 7 and Theorem 4.