Let $L$ be a $n \times n$ $Q$-matrix, $p_0$ probability vector such that $L(p_0) = p_0$ and $V\colon \Omega \rightarrow \mathbb{R}$ such that $V$ is constant in cylinders of the form $\{X_0 = c\}$. Then there are:

a) a unique positive function $u_V \colon \Omega \rightarrow \mathbb{R}$ constant equal to the value $u_V^i$ in each cylinder $\{X_0 = i\}$, $i \in \{1, \dots, n\}$;\

b) a unique probability vector $\mu_V \in \mathbb{R}^n$ such tha $\mu_V({i}) > 0$ and $\sum_{i=1}^n (u_V)_i (\mu_V)_i = 1$.

c) a real positive valeu $\lambda_V$ such that for any $s > 0$ $$e^{-s\lambda_V} u_V e^{s(L+V)} = u_V;$$ and given any vector $v \in \mathbb{R}^n$, $$\lim_{t \rightarrow +\infty} e^{-t\lambda_V} v e^{t(L+V)} = \left( \sum_{i=1}^n v_i (\mu)_i \right) u_V;$$

Moreover, given $s>0$, $e^{-s \lambda_V}e^{s(L+V)}\mu_V = \mu_V.$

Can someone give me a reference where I should find this result? Or maybe some proof.... Thanks in advance!

I'm looking for a proof (or a reference for it) of the following result:

Let $L$ be a $n \times n$ $Q$-matrix, $p_0$ probability vector such that $L(p_0) = p_0$ and $V\colon \Omega \rightarrow \mathbb{R}$ such that $V$ is constant in cylinders of the form $\{X_0 = c\}$. Then there are:

1. a unique positive function $u_V \colon \Omega \rightarrow \mathbb{R}$ constant equal to the value $u_V^i$ in each cylinder $\{X_0 = i\}$, $i \in \{1, \dots, n\}$;

2. a unique probability vector $\mu_V \in \mathbb{R}^n$ such tha $\mu_V({i}) > 0$ and $\sum_{i=1}^n (u_V)_i (\mu_V)_i = 1$;

3. a real positive value $\lambda_V$ such that for any $s > 0$ $$e^{-s\lambda_V} u_V e^{s(L+V)} = u_V;$$ and given any vector $v \in \mathbb{R}^n$, $$\lim_{t \rightarrow +\infty} e^{-t\lambda_V} v e^{t(L+V)} = \left( \sum_{i=1}^n v_i (\mu)_i \right) u_V.$$

Moreover, given $s>0$, $e^{-s \lambda_V}e^{s(L+V)}\mu_V = \mu_V.$

Thanks in advance!

Let $L$ be a $n \times n$ $Q$-matrix, $p_0$ probability vector such that $L(p_0) = p_0$ and $V\colon \Omega \rightarrow \mathbb{R}$ such that $V$ is constant in cylinders of the form $\{X_0 = c\}$. Then there are:

a) a unique positive function $u_V \colon \Omega \rightarrow \mathbb{R}$ constant equal to the value $u_V^i$ in each cylinder $\{X_0 = i\}$, $i \in \{1, \dots, n\}$;\

b) a unique probability vector $\mu_V \in \mathbb{R}^n$ such tha $\mu_V({i}) > 0$ and $\sum_{i=1}^n (u_V)_i (\mu_V)_i = 1$.

c) a real positive valeu $\lambda_V$ such that for any $s > 0$ $$e^{-s\lambda_V} u_V e^{s(L+V)} = u_V;$$ and given any vector $v \in \mathbb{R}^n$, $$\lim_{t \rightarrow +\infty} e^{-t\lambda_V} v e^{t(L+V)} = \left( \sum_{i=1}^n v_i (\mu)_i \right) u_V;$$

Moreover, given $s>0$, $e^{-s \lambda_V}e^{s(L+V)}\mu_V = \mu_V.$

Let $L$ be a $n \times n$ $Q$-matrix, $p_0$ probability vector such that $L(p_0) = p_0$ and $V\colon \Omega \rightarrow \mathbb{R}$ such that $V$ is constant in cylinders of the form $\{X_0 = c\}$. Then there are:

a) a unique positive function $u_V \colon \Omega \rightarrow \mathbb{R}$ constant equal to the value $u_V^i$ in each cylinder $\{X_0 = i\}$, $i \in \{1, \dots, n\}$;\

b) a unique probability vector $\mu_V \in \mathbb{R}^n$ such tha $\mu_V({i}) > 0$ and $\sum_{i=1}^n (u_V)_i (\mu_V)_i = 1$.

c) a real positive valeu $\lambda_V$ such that for any $s > 0$ $$e^{-s\lambda_V} u_V e^{s(L+V)} = u_V;$$ and given any vector $v \in \mathbb{R}^n$, $$\lim_{t \rightarrow +\infty} e^{-t\lambda_V} v e^{t(L+V)} = \left( \sum_{i=1}^n v_i (\mu)_i \right) u_V;$$

Moreover, given $s>0$, $e^{-s \lambda_V}e^{s(L+V)}\mu_V = \mu_V.$

Can someone give me a reference where I should find this result? Or maybe some proof.... Thanks in advance!