As we know, solution to $dX_t=\mu dt+\sigma dW_t$ is normal distributed and is light tailed; solution to $dX_t=\mu X_tdt+\sigma X_t dW_t$ is lognormal distributed and is heavy tailed. Is there any reference on discussion of criteria to identify the tail behavior from the driven SDE?

You want to consider: $$dx_t=f(x_t)dt + \sigma(x_t)dW_t$$ In dimension 1, you can express analytically the stationary probability density (if it exists) as: $$P(x)=\frac{1}{Z}\exp(\int^x \frac{f(u)}{\sigma(u)^2} du)$$ where $Z$ is a normalizing constant. Then you can express the various stationary moments using this formula and investigate the tail properties. 

