A limit theorem for singular stochastic differential equations

We study the weak limits of solutions to SDEs $dX_n(t)=a_n\bigl(X_n(t)\bigr)\,dt+dW(t),$ where the sequence $\{a_n\}$ converges in some sense to $(c_- 1\mkern-4.5mu\mathrm{l}_{x<0}+c_+ 1\mkern-4.5mu\mathrm{l}_{x>0})/x+γδ_0$. Here $δ_0$ 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.

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