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@George: thanks for your comments. I have a few question though, if I may. 1) Have you seen any literature on that topic? 2) convergent bounds that you provided are interesting, would you give a hint how to derive them? 3) for infinite paths an example that you gave for sure works. I think, it can be extended to the case when processes do not reach the same absorbing set (where kernels should coincide) but rather converge to that set and kernels are Feller. On the other hand, I thought that it may be can be further extended to the case when they just converge to the same invariant distribution
@Pascal: I've just accepted it - just wanted first to verify the answer by myself to be sure that I've understood in completely. I wonder if you can advise any references with exercises on the topic of $\phi$-irreducibility. Or maybe even references on the application of this theory to the analysis of the concrete models.
Thank you very much, Pascal. Correct me if I wrong: we can take $B = \{y:P(y,A) > 0\}$ and so $$ P^2 G(x,A) = \int\limits_E P(y,A)PG(x,dy) = \int\limits_B P(y,A)PG(x,dy) >0 $$ since $PG(x,B)>0$ for all $x\in E$ from the fact that $\phi(B)>0$. If I may ask - how did you get the intuition about this proof? I feel not very confident in dealing with closed sets and irreducibility so far, so in some problems for me it's hard to get the idea.
@Michael and @Pete I am bit confused with the use of Borel space for measurable space and not for a Borel subset of a Polish space. Is it a usual convention?
@ShawnD: well, to claim that something in mathematics doesn't need it proof is rather strong, isn't it? For example, it might happen that $v(x) = 1_{\mathbb R^n\setminus \{0\}}(x)$ even for a diffusion.
Well, it's not quite an SDE in the sense that the right-hand side does not depend on $X$, so you can write the explicit solution $$ X_t = X_0+\int\limits_0^t[a\sign W_s +b ]\mathrm dW_s. $$ On the other hand, the Kolmogorov equation is defined only for Markov processes, I guess. I am not sure that the process $X$ is Markov.
@Nathanael: could you refer me to any book (or maybe, lecture notes) on Dirichlet forms for discrete time Markov processes with general state spaces (I guess you call them simply Markov chains)?
@Nathanael: but this is only for the case $A=\mathscr X$, isn't it? In that case you obtain an answer just by writing $\Delta f(x)$ through the integral (or the sum in you example)
