Understanding the long time behaviour of solutions to ergodic stochastic differential equations is an important question with relevance in many field of applied mathematics and statistics. Hence, designing appropriate numerical algorithms that are able to capture such behaviour correctly is extremely important. A recently introduced framework [1,2,3] using backward error analysis allows us to characterise the bias with which one approximates the invariant measure (in the absence of the accept/reject correction). These ideas will be used to design numerical methods exploiting the variance reduction of recently introduced nonreversible Langevin samplers [4,5]. Finally if there is time we will discuss, how things ideas can be combined with the idea of Multilevel Monte Carlo  to produce unbiased estimates of ergodic averages without the need the of an accept-reject correction  and optimal computational cost.
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