Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|cite|improve this question
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

1 Answer 1

up vote 0 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.