Pathlevel solution of a rough differential equation ($y_t-y_s={\sum }_{1\le |I|\le \lfloor p\rfloor }(V_I\mathrm{id})(y_s)X_{s,t}^I+O(\omega (s,t{)}^{\theta })$)

A pathlevel solution is the ordinary state path satisfying a local rough Taylor expansion with a remainder of order $\theta >1$ in a control function for rough paths. It specifies the first level of the solution, rather than requiring the full lifted solution as part of the definition. With vector fields of regularity ${\mathrm{Lip}}^{\gamma }$, $\gamma >p$, the Davie lemma bounds discrete Euler errors uniformly; compactness and passage to the limit give such a path.