Behaviour of excessive functions along the trajectories of a Markov process

Properties of right and left (nonhomogeneous) Markov processes which have been constructed in [6] are studied in the paper. By a right (resp. left) process we mean a (nonhomogeneous) Markov process $x_t$ such that its transition and cotransition functions are a.s. rightcontinuous (resp. leftcontinuous) along trajectories. Due to Dynkin's results [2], excessive functions are a.s. rightcontinuous along trajectories of the right process. We consider excessive functions for the left process (or, which is the same, coexcessive functions for the right process). Let $h^t(x)$ be an (nonhomogeneous) excessive function for the left process $x_t$. It is proved that $h^t(x_t)$ does not have oscillatory discontinuity a.s. The set $\{s\colon h^t(x_t)$ is not leftcontinuous at $s\}$ is a.s. countable or finite and coincides a.s. with the set $\{s\colon (s,x_s)\in\Gamma\}$, where $\Gamma$ is an exceptional set depending on $h$. Conditions are introduced for the set $\Gamma$ to be empty.