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?

Thanks in advance!


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