# Ergodic Stochastic Differential Equations and Sampling: A numerical analysis perspective

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 [6] to produce unbiased estimates of ergodic averages without the need the of an accept-reject correction [7] and optimal computational cost.

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[2] A. Abdulle, G. Vilmart, and K. C. Zygalakis. High order numerical approximation of the invariant measure of ergodic sdes. SIAM J . Numer. Anal., 52(4):1600-1622, 2014.
[3] A. Abdulle, G. Vilmart, and K.C. Zygalakis, Long time accuracy of Lie-Trotter splitting methods for Langevin dynamics. SIAM J . Numer. Anal., 53(1):1-16, 2015.
[4] A. Duncan, G. A. Pavliotis and T. Lelievre, Variance Reduction using Nonreversible Langevin Samplers, J. Stat. Phys., 163(3):457-491, 2016.
[5] A. Duncan, G. A. Pavliotis and K. C. Zygalakis, Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation, arXiv:1701.04247
[6] M.B. Giles, Mutlilevel Monte Carlo methods, Acta Numerica, 24:259-328, 2015.
[7] L. Szpruch, S. Vollmer, K. C. Zygalakis and M. B. Giles, Multi Level Monte Carlo methods for a class of ergodic stochastic differential equations. arXiv:1605.01384.