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