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.