...we have $N$ states, with N very large. (In this application N is on the order of $10^{100000}$ or so.)

 

Given some state $k$, are there practical techniques to find out which of the two processes is more likely to have produced it?

If your question is whether there is a practical technique to answer your question given only this information then the answer is no. There are sparse matrix techniques that could be helpful when $N$ is reasonably sized but $N^2$ is too big, but when $N$ itself is so unwieldy, you will have to break open the transition matrix black box. In other words you will probably have to use some of the details of your simulation program to answer this question.

...we have $N$ states, with N very large. (In this application N is on the order of $10^{100000}$ or so.)

Given some state $k$, are there practical techniques to find out which of the two processes is more likely to have produced it?

If your question is whether there is a practical technique to answer your question given only this information then the answer is no. There are sparse matrix techniques that could be helpful when $N$ is reasonably sized but $N^2$ is too big, but when $N$ itself is so unwieldy, you will have to break open the transition matrix black box. In other words you will probably have to use some of the details of your simulation program to answer this question.