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A control is a continuous nonnegative function on ordered pairs of times, zero on the diagonal and superadditive as displayed. It measures the size of increments of a rough path, typically through $|X_{s,t}^{(k)}|\le \omega (s,t{)}^{k/p}$. Continuous finite variation gives suitable controls, which let local estimates be stated without tying them to a particular time parametrization.

A rough differential equation interprets a differential equation using the higher-order data of its driving rough path. For $2<p<3$, its local expansion includes $V_i(Y_s)X_{s,t}^i+D{V}_j(Y_s)V_i(Y_s){\mathbb{X}}_{s,t}^{ij}$, with a remainder controlled at order greater than one. For a smooth driver it is the usual controlled ordinary differential equation; for enhanced Brownian motion it is the corresponding Stratonovich equation.

The product of two coordinate iterated integrals of a path is the sum over order-preserving interleavings of their integration variables. Split the product of the two ordered simplices according to the joint order of all variables. This identity explains the algebraic constraints on a geometric rough path and the cancellation of low-degree terms in the composition of its local Euler maps.

For a Borel probability measure on a Polish space and a continuous map to another Polish space, the image measure has the displayed support. Every neighborhood of an image of a support point has an open inverse image of positive measure. Conversely, the closed image closure has full image measure because the original support has full measure. This transfers a support description for a driving rough path to the solution of a rough differential equation.