Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 32 / 1

Put $V={\mathbb{R}}^d$ and write $a^{(k)}$ for the degree-$k$ component of a tensor. We use the truncated tensor algebraHere $V^{\otimes 0}=\mathbb{R}$, and products of degree above $N$ are discarded. The answers to the three printed parts follow with these conventions; the middle part is missing from the converted TeX.

In the truncated tensor algebra, let ${\mathfrak{g}}^N(V)$ be the linear span of all iterated Lie brackets of elements of $V$ of length at most $N$, with bracket $[a,b]=a\otimes b-b\otimes a$. Brackets of length greater than $N$ vanish. The free step-N nilpotent Lie group isThe exponential and logarithm are the finite seriesIts multiplication is the tensor multiplication above, its identity is $e=(1,0,\dots ,0)$, and its inverse isThe truncated Baker--Campbell--Hausdorff formula shows that this multiplication stays in the exponential of the Lie algebra. This is the connected simply connected Lie group of the free step-$N$ nilpotent Lie algebra, rather than a discrete free nilpotent group.

Give the first layer $V$ its usual Euclidean norm. An absolutely continuous curve $\gamma$ in $G^N(V)$ is horizontal iffor an absolutely continuous control $x$ in $V$. In the truncated tensor algebra this equation is ${\dot{\gamma }}_t={\gamma }_t\otimes {\dot{x}}_t$, truncated at degree $N$. Its horizontal length is $\int |{\dot{x}}_t|dt$. The Carnot-Carathéodory distance isThe first-layer Lie brackets span the full Lie algebra, so horizontal curves reach every endpoint. Thus this infimum is finite; it is a left-invariant distance inducing the usual manifold topology. Dilations ${\delta }_{\lambda }(a{)}^{(k)}={\lambda }^k{a}^{(k)}$ give $d_{CC}({\delta }_{\lambda }g,{\delta }_{\lambda }h)=\lambda d_{CC}(g,h)$ for $\lambda >0$. Equivalently $d_{CC}(e,g)$ is the least length of a bounded variation control whose step-$N$ path signature is $g$.

For $p\ge 1$ set $N=\lfloor p\rfloor$. A weak geometric p-rough path is a continuous path $\mathbf{X}:[0,T]\to G^N(V)$, usually based at $e$, withIts increments ${\mathbf{X}}_{s,t}={\mathbf{X}}_s^{-1}\otimes {\mathbf{X}}_t$ obey ${\mathbf{X}}_{s,u}={\mathbf{X}}_{s,t}\otimes {\mathbf{X}}_{t,u}$. Equivalently their degree-$k$ tensor components have finite $p/k$-variation and satisfy the path signature algebraic constraints. This is a group-valued definition; being a limit of smooth path signatures in the same $p$-variation topology is the stronger definition of a geometric p-rough path.

For $V={\mathbb{R}}^2$ with basis $e_1,e_2$, the step-two Lie algebra has basis $e_1,e_2,[e_1,e_2]$ and its last element is central. WriteIts first level is $v=(x,y)$ and its second level isThe step-two multiplication givesConsequently the mapis an isomorphism onto the real Heisenberg group. The shift by $xy/2$ converts exponential coordinates into the matrix coordinates used for that group. In exponential coordinates the left-invariant horizontal vector fields areTaking $X,Y$ orthonormal gives exactly the Carnot-Carathéodory distance inherited from $G^2({\mathbb{R}}^2)$.

Euclidean Lipschitz continuity does not imply Carnot-Carathéodory Lipschitz continuity. In the exponential coordinates above take the vertical path $\gamma (t)=(0,0,t)$. It is Euclidean Lipschitz continuous, butIndeed left invariance reduces to the central increment, graded dilation gives the square-root factor, and reversal changes the sign without changing the distance.

One can also see the scale directly. A horizontal curve starting at the identity satisfies $\dot{z}=(x\dot{y}-y\dot{x})/2$. If it ends on the vertical axis, its planar projection is a closed loop andFor a projection of length $L$ starting at zero, $\sup |(x,y)|\le L$, so $|z|\le L^2/2$. Thus reaching a central displacement $z$ requires length at least $\sqrt{2|z|}$, while a square loop with the appropriate signed area gives length at most $4\sqrt{|z|}$. In particular $d_{CC}(\gamma (s),\gamma (t))/|t-s|$ is unbounded as $t\to s$. Vertical displacement records area, which scales quadratically with horizontal length.