Let $W_t$ be a Brownian motion defined on some probability space and let $X$ be the (strong) solution to the SDE:
\begin{equation} dX_t = b(t, X)dt + \sigma(t, X)dW_t, \end{equation} with $X(0) = X_0$. I am interested in the limit (in distribution)
\begin{equation} \lim_{t\rightarrow\infty} X_t. \end{equation} In particular, I would like to know when it is well-defined and, when it is defined, what its distribution is.
As an example, when $b(t, x) = 0$, $\sigma(t, x) = 1$, the solution is Brownian motion, which does not have a limit in distribution as $t\rightarrow\infty$. However, if instead $b(t, x) = -x$ we have a particular case of the Ornstein-Uhlenbeck process which has a limiting distribution.
Can anyone recommend a good text which covers this subject?