The Ornstein-Uhlenbeck process has the form $$d X_{t}=-\kappa X_t d t+\sigma d W_{t}$$ for $\kappa$ and $\sigma >0$, has the solution $X_t$ is normal with parameters \begin{aligned} &E[X(t)]=e^{-\kappa\left(t-t_{0}\right)} E\left[x_{0}\right]\\ &\operatorname{Var}[X(t)]=\frac{\sigma^{2}}{2 \kappa}\left(1-e^{-2 \kappa\left(t-t_{0}\right)}\right)+e^{-2 \kappa\left(t-t_{0}\right)} \operatorname{Var}\left[x_{0}\right] \end{aligned}
I would like to know how a process of the form $$d X_{t}=-\kappa X_t^n d t+\sigma d W_{t}$$ for $n\in \mathbb N$ can be solved. Do these processes have a name`?
I can tell what the stationary distribution of such processes look like, since they need to solve the Fokker Plank equation, but not the solution.