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My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time and discrete state Markov chain with transition matrix; if $\Omega$ is a continuous space, it is a discrete-time and continuous state Markov process).

Given the transition rule $P(x_{t+1},x_{t})$,if every state satisfies below three conditions:(1)positive recurrence (2)communication (3)aperiodical.Is there an unique stationary measure ? i.e. $\pi(y)=\int_{\Omega}p(x,y)\pi(x)dx$ has nontrivial solution $\pi(x)$ as an invariant measure.

In the discrete-time and discrete state Markov chain with transition matrix, the answer is sure: there must exist a stationary distribution once above conditions are satisfied as the statement in classical textbook (e.g. Ross) .

I'm not sure above claims is right or not for the discrete-time and continuous state Markov process satisying above three conditions ?

Could some guys help me verify this judegement or recommend some related books or paper?

I just read the Harris' paper "The existence of stationary measures for certain Markov process"(http://projecteuclid.org/download/pdf_1/euclid.bsmsp/1200502009) ,this paper focus on the discrete-time Markov process with general transition law.Theroem 1 seems to use more relaxed conditions:only requiring positive reccurence(condition c in his paper).

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  • $\begingroup$ Maybe you can be more explicit about your definitions. Do you really want your state space to be completely general (e.g. any measurable space) or are you willing for it to be, e.g., standard Borel? And what do you want "positive recurrence" and "communication" to mean, exactly? If you mean "for all $x$ we have $E_x[\tau_x] < \infty$" and "for all $x,y$ we have $P_x(\tau_y < \infty) = 1$" then these are extremely strong assumptions that are probably not satisfied by very many useful examples. $\endgroup$ – Nate Eldredge Jul 18 '15 at 23:43
  • $\begingroup$ For example, an iid sequence drawn from a continuous distribution does not satisfy them. Usually when dealing with continuous-state models, you have to introduce a topology on the state space to be able to say anything useful. $\endgroup$ – Nate Eldredge Jul 18 '15 at 23:44
  • $\begingroup$ Yes , the state space I care is just an interval $[a,b]$. The meaning of "recurrence" here, I use the Harris's definition:Suppose the Borel set generated by $[a,b]$ is B,and Lebegue measure is L, for $\forall E \in B,\forall x_{0} \in [a,b]$ and $L(E)>0$, there will be $P(x_{t} \in E infinitely many|x_{0})=1$,then I call this process is positive recurrence. $\endgroup$ – Galor Jul 19 '15 at 2:31

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