Asymptotics for the normalized error of the Ninomiya-Victoir scheme

In a previous work, we proved strong convergence with order $1/2$ of the Ninomiya-Victoir scheme $X^{NV,\eta}$ with time step $T/N$ to the solution $X$ of the limiting SDE. In this paper we check that the normalized error defined by $\sqrt{N}\left(X - X^{NV,\eta}\right)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher random variables $\eta$. This result can be seen as a first step to adapt to the Ninomiya-Victoir scheme the central limit theorem of Lindeberg Feller type, derived by M. Ben Alaya and A. Kebaier for the multilevel Monte Carlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. This suggests that the rate of convergence is greater than $1/2$ in this case and we actually prove strong convergence with order $1$.