A limit theorem for singular stochastic differential equations

We study the weak limits of solutions to SDEs dX(n)(t) = a(n)(X-n(t)) dt + dW(t), where the sequence {a(n)} converges in some sense to (c-1 (x< 0)+ c + 1 (x> 0))/x + gamma delta(0) Here delta(0) is the Dirac delta function concentrated at zero. A limit of {X-n} may be a Bessel process, a skew Bessel process, or a mixture of Bessel processes.