Consider an SDE (say scalar valued, with some initial value)
\begin{equation*} dX_t=b(X_t)dt+\sigma(X_t)dW_t. \end{equation*}
Fix a time step $\Delta t>0$, and consider its Euler-Maruyama approximation for $j=1,2,\ldots$
\begin{equation*} X^{\Delta t}_{j+1}=X^{\Delta t}_j+b(X^{\Delta t}_j)\Delta t + \sigma (X^{\Delta t}_j) \Delta W_j \end{equation*}
(where $\Delta W_j=W_{j+1}-W_j$). Of course we can send the time step $\Delta t$ to zero and consider the strong and weak error of this numerical method. However I'm looking for a reference which considers the behaviour of the distribution of $X^{\Delta t}_j$ as $j \to \infty$ ( for fixed $\Delta t$ ) - can someone please provide a reference or some insight on this? Of course we will probably have to start by assuming some ergodic properties of the exact process $X_t$.
What is this type of analysis of a numerical solution of an SDE called? Stability?