Pathwise estimates for effective dynamics: The case of nonlinear vectorial reaction coordinates

Effective dynamics using conditional expectation was proposed in [F. Legoll and T. Lelièvre, Nonlinearity, 23 (2010), pp. 2131-2163] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [F. Legoll, T. Lelièvre, and S. Olla, Stochastic Process. Appl., 127 (2017), pp. 2841-2863], in this paper we obtain pathwise estimates of effective dynamics for both nonlinear and vector-valued reaction coordinate functions.