Are these two some kinds of generalized Ornstein–Uhlenbeck processes?

Question

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? What are their generators like? How are their attraction regions like and decided?

Thanks in advance!

Answer 1

I've never seen these processes, but you can quite easily come up with generator since $$ \mathrm dX_t = a(X_t)\mathrm dt + \mathrm dW_t $$ has a generator $$ \mathscr Af(x) = a(x)f'(x)+\frac12 f''(x) $$