As we know, to prove the convergence of stochastic process, we could either show the convergence of finite dimensional distribution and tightness of the process, or use techniques of martingale problems. What about the following Markov process:

$L=\frac{1}{2}p(1-p)\frac{d^{2}}{dp^{2}}-\frac{\theta}{2}p\frac{d}{dp}+\log(\theta) p(1-p)(2p-1)\frac{d}{dp}, p\in[0,1]$

We can see that the generator explodes when $\theta\rightarrow0$. How can we find the limit of this process as $\theta\rightarrow0$. Apparently, the techniques of martingale problems are not applicable here!