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).

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).

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).

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).