# Attractors and solutions to these generalized Ornstein–Uhlenbeck processes?

This is a question about generalized Ornstein–Uhlenbeck processes I asked on MSE, but I haven't received replies about their attractors and solutions yet. So I would appreciate if someone could give me at least some hints. Thanks a lot!

An Ornstein–Uhlenbeck process is $$d X_t = (\mu - X_t) dt + d W_t$$

We try to build a model using some generalized Ornstein–Uhlenbeck processes.

1. The first one is $$d X_t = \exp(-|X_t- \mu|) (\mu - X_t) dt + d W_t$$ where we hope $\exp(-|X_t- \mu|)$ will reduce the speed of $X_t$ approaching $\mu$, as $X_t$ comes closer to $\mu$.

2. Furthermore, since an O-U sde has a attractor $\mu$, we tempt to generalize the above sde to have more than one attractors $$d X_t = \sum_{i=1}^3 \exp(-|X_t- \mu_i|) (\mu_i - X_t) dt + d W_t$$

I have little idea about these two generalized Ornstein–Uhlenbeck processes. So may I ask here if there are some references on them?

Do they have weak or strong solutions and how to determine their solutions?

How are their attraction regions like and decided?