Ergodicity of diffusion and temporal uniformity of diffusion approximation

Let {XN (t), t ≧ 0}, N = 1, 2, … be a sequence of continuous-parameter Markov processes, and let TN (t)f(x) = Ex [f(XN (t))]. Suppose that limN→∞ T N (t)f(x)= T(t)f(x), and that convergence is uniform over x and over t ∈ [0, K] for all K < ∞. When is convergence uniform over t ∈ [0, ∞)? Questions of this type are considered under the auxiliary condition that T(t)f(x) converges uniformly over x as t → ∞. A criterion for such ergodicity is given for semigroups T(t) associated with one-dimensional diffusions. The theory is illustrated by applications to genetic models.