**(Edited)** I met a univariate Ornstein-Uhlenbeck type process but with self soft-thresholding:
$$
dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\mu}\big]_+ dt + \sigma dB(t), \quad X(0)=0,
$$
where $B(t)$ is the standard Brownian motion, and $c,c_1,\sigma>0$ are constants, and $[\cdot]_+$ is a soft-thresholding operator with $[a]_+=a$ if $a> 0$ and $[a]_+=0$ if $a\leq 0$.

**My Question**: does the stationary distribution exist for the SDE above, for different $\mu\in[0,1]$? I suspect that there is a regime switching phenomenon at around $\mu=1/2$...

Any comment will be much appreciated!