Joint with Carlo Bellingeri, Peter Friz and Sylvie Paycha
With thanks to Terry Lyons
It has become common folklore in rough path analysis that spaces of weakly geometric, quasi geometric or branched rough paths do not carry any meaningful notion of vector field structure, that their sum can in general only made sense of in terms of a non-unique Lyons-Victoir coupling. However, the situation drastically changes if one restricts to the at first sight seemingly paradox concept of smooth rough paths. We briefly present the Lie group theoretic aspect behind this surprising construction, show how it emerges from an 1998 statement of Lyons and explain how this sum is connected to the Bruned-Chevyrev-Friz-P. renormalization of rough paths applied to smooth rough paths.