= Pathlevel solution of a rough differential equation
{title2=$y_t-y_s=\sum_{1\leq|I|\leq\lfloor p\rfloor}(V_I\operatorname{id})(y_s)X^I_{s,t}+O(\omega(s,t)^\theta)$}
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 $\operatorname{Lip}^\gamma$, $\gamma>p$, the <Davie lemma> bounds discrete Euler errors uniformly; compactness and passage to the limit give such a path.
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