Thank you! I ran some simulations. For $\gamma>1/2$, the behavior of $X$ does look like $\sigma B$. For $\gamma<1/2$, $X$ explodes. However, rather than oscillating between positive and negative values, $X$ explodes toward one of the directions. Some simulation paths explode toward $+\infty$, some toward $-\infty$, but they do not oscillate.

Thank you. Suppose I consider the following process instead:$$ dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\gamma}\big]_+ dt + \sigma dB(t), \quad X(0)=0, $$ where $c_1>0$, I wonder if anything can be said about the behavior of $X(t)$ with respect to different value of $0 < \gamma\leq 1$?