In a related question I understood how my current problem is related to speeding-up Monte Carlo sampling for same family of measures. The literature is vast and the strategies are multiple. Since I recently refreshed some SDE theory, my question is:

is it a good idea to go deeper in (numerical) SDEs, having as ultimate goal the improvement of Monte Carlo algorithms? Is there a standard reference for the field?

More context follows. For instance, I know that if a measure $\mu$ is ergodic for an SDE process $X_t$, an approximation of $\mu$ can be obtained by long-time simulation of $X_t$, but this is usually not feasible because discretization breaks such a property. But it's not a dead end: the numerical scheme can be used as *proposal* for a Metropolis-Hasting algorithm, with final performance sometimes better than standard methods (this is the MALA algorithm; here there is an answer with a nice explanation).

I also found some recent papers on the subject, written by mathematicians/statisticians/cs people. What said gives me motivation and a sense of challenge, suggesting that more can be done and that this area is active.

I have a background in mathematics and recently started a PhD in a Computer Science Department. I spoke with my supervisor and he gave me a good degree of freedom: roughly most of the time is spent between programming and more-concrete-algorithms, but a decent rest is left free to my choice, therefore my question. That topic would be my *bonus*, while working at the same time on topics more known by my supervisor. As always, thank you very much for your attention.