So my question is motivated by the following:
Note that the ODE $$ dy_t = 2sgn(y_t)\sqrt{|y_t|}$$ $$y_0 = 0$$$$ has no unique solution. However, consider the SDE as follows:
$$ dy_t = 2sgn(y_t)\sqrt{|y_t|} + \sigma dW_t$$ $$ y_0 = 0$$
What happens to this SDE as $\sigma \rightarrow 0$? Does this have a unique solution? My intuitive guess is that it is going to converge weakly to
$$(Ax^2 -x^2(1-A))\chi_{[0,\infty]}$$ where the law of $A$ is $\frac{(\delta_0 + \delta_1 )}{2}$. So, essentially the process that is 0 for t < 0, and is either $x^2$ or its negative with equal probability thereafter. But I have no idea how to prove this. Any thoughts?