Usually an SDE $dX_t = f(X_t,t) + g(X_t,t)dW_t$ is formulated in 1-dimensional time domain, $t\in \mathbb{R}$. However, in principle, the time could be also a subset of $\mathbb{R}^2$, $t=(t_1,t_2) \in \mathbb{R}^2$. Does someone know some reference studying such a process?
In particular, let's say the process is an OU in both $t_1$ and $t_2$ directions. Can we say something about the limiting process?