Before revision:  2013, arxiv:1305.6765 suggests to work with the rescaled variable X^{\varepsilon}:=\varepsilon^{1/(1-\gamma)} X: while allowing to turn a spatial asymptotic problem into a  fixed terminal point, small-\varepsilon problem, the process X^{\varepsilon} satisfies a SDE in Wentzell--Freidlin form (i.e. with driving noise \varepsilon dB).
After revision:  2013, arxiv:1305.6765 suggests to work with the rescaled variable X^{\varepsilon}:=\varepsilon^{1/(1-\gamma)} X: while allowing to turn a space asymptotic problem into a fixed terminal point, small-\varepsilon problem, the process X^{\varepsilon} satisfies a SDE in Wentzell--Freidlin form (i.e. with driving noise \varepsilon dB).