# Why the term "geometric" rough path?

A "geometric" rough path is a rough path such that $Sym(\mathbb{X}_{s,t})=\frac{1}{2}X_{s,t}\otimes X_{s,t}$. For example the Ito rough path is not geometric because $Sym(\mathbb{X}_{s,t})=\frac{1}{2}X_{s,t}\otimes X_{s,t}-\frac{1}{2}I(t-s)$ but the Stratonovich rough path is geometric.

Why the term "geometric" here? Is there some intuition I'm missing? When I think geometric, I think of geometric Brownian motion, (http://en.wikipedia.org/wiki/Geometric_Brownian_motion) or geometric series, something with an exponent. I don't see that here.

• The definition I've seen goes like this: a smooth path has a canonical enhancement into a "smooth rough path". Then the geometric rough paths are the closure of the smooth rough paths, in the appropriate $p$-variation topology. So it could be that geometric rough paths have, in some sense, an enhancement that is geometrically reasonable, rather than arbitrary. May 27, 2015 at 3:44
• Thanks for your response. I guess I don't understand, why is that condition geometrically reasonable?
– user69208
May 27, 2015 at 3:46
• I don't really know either. Another guess is the fact that the Stratonovich integral is more "geometrically reasonable" than Ito: if you have a submanifold $M \subset \mathbb{R}^n$ and a vector field $V$ which is tangent to $M$, the process $dX_t = V(X_t) \cdot \delta B_t$ (Stratonovich) will lie in $M$, but $dX_t = V(X_t) \cdot dB_t$ (Ito) will not. I am not sure but there may be some similar property of geometric rough paths. Anyway, I think one should think of "geometric" meaning more like "differential geometry" than "geometric series". May 27, 2015 at 3:53

Geometric rough paths have the property that if you want to solve an equation with values in a manifold, choose a coordinate chart, and write in local coordinates $$dY^i = V_0^i(Y)\,dt + \sum_j V_j^i(Y)\,dX_j$$ for some vector fields $V_i$ (with the obvious abuse of notation that the solution actually depends on the choice of $\mathbb{X}$, not just on $X$), then the solution does not depend on the choice of chart. I believe that this is the reason why this terminology was chosen.