  This work is concerned with model reduction of stochastic differential
equations and builds on the idea of replacing drift and noise coefficients of
preselected relevant, e.g. slow variables by their conditional expectations. We
extend recent results by Legoll & Leli\`evre [Nonlinearity 23, 2131, 2010] and
Duong et al. [Nonlinearity 31, 4517, 2018] on effective reversible dynamics by
conditional expectations to the setting of general non-reversible processes
with non-constant diffusion coefficient. We prove relative entropy and
Wasserstein error estimates for the difference between the time marginals of
the effective and original dynamics as well as an entropy error bound for the
corresponding path space measures. A comparison with the averaging principle
for systems with time-scale separation reveals that, unlike in the reversible
setting, the effective dynamics for a non-reversible system need not agree with
the averaged equations. We present a thorough comparison for the
Ornstein-Uhlenbeck process and make a conjecture about necessary and sufficient
conditions for when averaged and effective dynamics agree for nonlinear
non-reversible processes. The theoretical results are illustrated with suitable
numerical examples.
