# Probability approximation: monte carlo VS sde

I have a probability measure $$\mu$$ (say, in $$\mathbb{R}^{d}$$, with density) and I want to approximate it numerically. Today I noticed that my measure is ergotic for a certain Stochastic Differential Equation. If we call $$x$$ our starting point (asymptotically not relevant), to sample from $$\mu$$ I see two alternatives:

1. run the usual Metropolis-Hasting chain (MCMC);
2. simulate a SDE path starting from $$x$$ (e.g. via Euler's discretization), then stop at large times;

I am right, or did I misunderstand the strategy number 2? How is the efficiency of these two algorithms compared, in general?

My book on Monte Carlo methods does not consider SDEs, while mine on SDEs does ignore the Metropolis algorithm. Good rigorous references (as well as papers) are highly appreciated. Thanks in advance.