A limit theorem for singular stochastic differential equations

We study the weak limits of solutions to SDEs [dX_n(t)=a_nbigl(X_n(t)bigr),dt+dW(t),] where the sequence ${a_n}$ converges in some sense to $(c_- 1mkern-4.5mumathrm{l}_{x<0}+c_+ 1mkern-4.5mumathrm{l}_{x>0})/x+gammadelta_0$. 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.