  In this paper we study a type of two-player zero-sum stochastic differential
games with ergodic payoff, in which the diffusion system does not need to be
non-degenerate. We first show the existence of a viscosity solution of the
associated ergodic Hamilton-Jacobi-Bellman-Isaacs equation under the
dissipativity condition. With the help of this viscosity solution, we then give
the estimates for the upper and lower ergodic value functions by constructing a
series of non-degenerate approximating processes and using the sup- and
inf-convolution techniques. Finally, we obtain the existence of a value under
the Isaacs condition and its representation formulae. In addition, we apply our
results to study a type of pollution accumulation problems with the long-run
average social welfare.
