Formulas for eigen-functions and eigen-measures associated with Markov process

Let ($x_t$, $\zeta$, $\mathscr M_t$, $\mathbf P_x$) be a strong Markov process, $\tau_1$ a Markov time, $a=a_1+ia_2$ a complex number, $\mathbf M_xe^{a_1\tau_1}<\infty$. We can consider two operators with the kernel
$$ q_a(x,\Gamma)=\mathbf M_xe^{a_1\tau_1}\chi_\Gamma(x_{\tau_1}) $$
($\chi_\Gamma$ stands for the indicator function of the set $\Gamma$, $\mathbf M_x$ for the expectation corresponding to the probability measure $\mathbf P_x$), one acting upon functions, the other upon measures. Let us call $a$-eigen-functions (measures) eigen-function (measures) of the semi-group generator connected with the Markov process that correspond to the eigen-value $-a$. For certain classes of Markov times, there is a one-to-one correspondence between $a$-eigen-functions (measures) and eigen-functions (measures) of $q_a$ with the eigen-value 1. As for functions, this correspondence is expressed in an obvious way, but for measures the following holds: If $\nu=\nu q_a$, then $\mu$ is an $a$-eigen-measure,
$$ \mu(\Gamma)=\int\nu\,(dx)\mathbf M_x\int_0^{\tau_1}e^{at}\chi_\Gamma(x_t)\,dt $$
in continuous parameter case; for Markov chains the inner integral is replaced by a sum.
This relation is a generalization of a formula for invariant measures (i.e. $a=0$) which was introduced in many papers ([2]–[8]).
The class of admissible Markov times includes times when a motion cycle between two disjoint sets ends; for these Markov times, the whole construction is a generalization of an approach to eigen-functions of a differential operator based on Schwartz alternating method.
The results concerning the correspondence between $a$-eigen functions (measures) and eigen-functions (measures) of $q_a$ can be applied to investigate the asymptotical behaviour of eigen-values and eigen-functions of a differential operator with small parameter (cf. [13]).