Representation of a semi-Marcov process as a time changed Markov process

Let $(P_x)$ be a $\lambda$-continuous and $\lambda$-regular semi-Markov process on a complete separable locally compact metric space $X$ and let $A_\lambda=A_0+A_\lambda 1\cdot I$ be the $\lambda$-characteristical operator of the process. If $\lambda R_\lambda\varphi\to\varphi$ ($\lambda\to\infty$) uniformly on $X$ where $\varphi\in C_0$ and $R_\lambda$ is the resolvent operator of the process and if $A_\lambda 1$ is continuous negative function on $X$, $A_{0+}1=0$, $A_\lambda 1\to -\infty$ ($\lambda\to\infty$) then for all $\lambda_0>0$ there exists a Markov process which differs from $(P_x)$ by random change of time only. The operator $\bar A=-(\lambda_0/A_{\lambda_0})$ is an infinitesimal operator of the Markov process and
$$a_t(\lambda)=\lambda_0\int_0^t\frac{A_{\lambda}1}{A_{\lambda_0}1}\circ\pi_s\,ds\qquad(\lambda>0)$$
($\pi_s(\xi)=\xi(s)$, $\xi$ is a trajectory of the process) is a Laplace family of additive functionals which determines the random change of time.