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1. Introduction. The problem of deriving suitable hypotheses from which one can conclude the weak convergence of a family of diffusion processes xn(t, co) to a limiting diffusion process x(t, co) as «->oo has attracted much attention in recent years. One popular approach is to study the asymptotic behavior of the diffusion coefficients. Specifically let us assume that {xn(t, co), 1 ^n< + 00} and x(t, co) are one-dimensional, strong Markov processes with continuous paths and stationary transition probabilities. Assume further that the infinitesimal generators Gn and G of the corresponding semigroups Tn(t)f(x) = Eœf(xn(t,oo)) and T(t)f(x) = Exf(x(t9oo)) are classical second order differential operators of the form : Gnf(x) = an(x)f\x) + bn(x)f'(x), an(x) > 0, (1) Gf(x) = a(x)f"(x) + b{x)f\x\ a(x) > 0.

A convergent family of diffusion processes whose diffusion coefficients diverge