  We develop the algebraic theory of rough path translation. Particular
attention is given to the case of branched rough paths, whose underlying
algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model
case of a regularity structure in the sense of Hairer. Pre-Lie structures are
seen to play a fundamental rule which allow a direct understanding of the
translated (i.e. renormalized) equation under consideration. This construction
is also novel with regard to the algebraic renormalization theory for
regularity structures due to Bruned--Hairer--Zambotti (2016), the links with
which are discussed in detail.
