Large deviations for excursions of non-homogeneous Markov processes

In this paper, the large deviations at the trajectory level for ergodic Markov processes are studied. These processes take values in the non-negative quadrant of the two-dimensional lattice and are concentrated on step-wise functions. The rates of jumps towards the axes (downward jumps) depend on the position of the process - the higher the position, the greater the rate. The rates of jumps going in the same direction as the axes (upward jumps) are constants. Therefore the processes are ergodic. The large deviations are studied under equal scalings of both space and time. The scaled versions of the processes converge to 0. The main result is that the probabilities of excursions far from 0 tend to 0 exponentially fast with an exponent proportional to the square of the scaling parameter. The proportionality coefficient is an integral of a linear combination of path components. A rate function of the large deviation principle is calculated for continuous functions only.