  In this paper, the large deviations on trajectory level for ergodic Markov
processes are studied. These processes take values in the non-negative quadrant
of the two dimension lattice and are concentrated on step-wise functions. The
rates of jumps towards the axes (jump down) 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 (jump up) 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 long excursions out of 0 tend to 0
exponentially fast with an exponent proportional to the square of the scaling
parameter. A 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.
