Rough path

A rough path augments a path with compatible higher-order iterated-integral data. The extra data records effects, such as signed area, that are not continuously determined by the first-level path in the uniform topology. A rough differential equation uses this enhancement to recover a continuous solution map for irregular drivers.

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.

A pathlevel solution is the ordinary state path satisfying a local rough Taylor expansion with a remainder of order $\theta >1$ in a control function for rough paths. It specifies the first level of the solution, rather than requiring the full lifted solution as part of the definition. With vector fields of regularity ${\mathrm{Lip}}^{\gamma }$, $\gamma >p$, the Davie lemma bounds discrete Euler errors uniformly; compactness and passage to the limit give such a path.

For a finite time grid, suppose a nonnegative error is zero on neighboring points and obeys $a_{s,t}\le a_{s,u}+a_{u,t}+L\omega (s,t{)}^{1/p}{a}_{s,u}+A\omega (s,t{)}^{\theta }$. For a control function for rough paths and $\theta >1$, sufficiently small control intervals have the displayed bound with constants independent of the grid. Splitting near half the control size makes the higher-order errors contract, and handles a crossing single grid interval by its zero error. Applying this to local Euler composition errors yields compactness of approximate pathlevel solutions.

For sufficiently regular vector fields, for example bounded fields with bounded derivatives through order three when $2<p<3$, the solution map of a rough differential equation exists uniquely and is locally continuous in the rough path metric. Ordinary differential equations driven by smooth approximations therefore converge whenever their enhanced drivers converge. The theorem makes the enhancement, rather than just the original path, the correct input for a continuous solution map.

The Stratonovich enhancement of Brownian motion is a geometric $p$-rough path for every $p>2$. Its second level consists of Lévy area and the prescribed symmetric tensor. Its increments satisfy the Chen identity and have finite Hölder rough-path norms at every exponent below $1/2$.

For every $2<p<3$ and every positive radius, the canonical Brownian enhancement has positive probability of lying in the displayed rough path metric ball. This controls both displacement and area. Together with Cameron-Martin quasi-invariance of enhanced Brownian motion, it puts every lifted Cameron-Martin control in the support of enhanced Brownian motion.

For $2<p<3$, the support of a measure of enhanced Brownian motion is the rough-path closure of path signatures of paths in the Cameron-Martin space of Wiener measure. Polygonal convergence gives one inclusion. For the reverse inclusion, decompose Brownian motion into its polygonal interpolation and independent pinned bridges. The bridge enhancement becomes small in probability as the mesh decreases, while every finite-dimensional neighborhood of the polygonal vertices has positive Gaussian probability. Rough path translation then gives positive probability in every ball around a prescribed control path signature.

For deterministic nested partitions with mesh tending to zero, lift the polygonal interpolation of Brownian motion by its ordinary iterated integrals. These path signatures converge almost surely to enhanced Brownian motion in every Hölder rough-path metric of exponent below $1/2$, hence in $p$-variation for $p>2$. Conditioning on the partition values gives a nested martingale for the first level and the off-diagonal second levels. Moment bounds and interpolation upgrade pointwise convergence to convergence in the rough path metric.

Condition on the Brownian values at a dyadic mesh. First-level increments and antisymmetric area increments of its polygonal lift are conditional expectations of the corresponding Brownian increments. Conditional Jensen inequalities and the homogeneous degree-two norm give the displayed bound. The Doob maximal inequality makes the right-hand side uniformly finite almost surely when the Brownian Hölder norm has sufficiently many moments. Dense-time martingale convergence and equicontinuity then yield simultaneous convergence at every time.

The path signature collects the ordered iterated Riemann-Stieltjes integrals of a continuous path of bounded variation. Its truncation belongs to the free step-N nilpotent Lie group. It multiplies under concatenation by the Chen identity, is independent of orientation-preserving reparametrisation of the whole path, and records more than its endpoint displacement.

For a continuous path of bounded variation, integrate successively over the ordered simplex using Riemann-Stieltjes integrals. These integrals form the levels of its path signature. Appending the final integration variable gives the differential recursion $d{S}_t^{(k)}=S_t^{(k-1)}\otimes d{x}_t$.

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.

Partition each ordered integration simplex according to the number of times before the concatenation point. The integral over each resulting region factors into an initial iterated integral and a final iterated integral. Summing these factorizations proves the identity in the truncated tensor algebra.

This inhomogeneous variation metric compares the tensor increments level by level. A common alternative is the Hölder metric ${\rho }_{\alpha }={\max }_k{\sup }_{s<t}|X_{s,t}^{(k)}-Y_{s,t}^{(k)}|/|t-s{|}^{k\alpha }$. On a finite interval convergence in this metric implies $p$-variation convergence whenever $p\alpha \ge 1$. Both topologies control the iterated integrals needed by the universal limit theorem.

A geometric $p$-rough path is a limit, in the rough path metric, of path signatures of sufficiently regular paths. It is a weak geometric p-rough path. At the same exponent these definitions need not coincide; weak geometric paths can be approximated after slightly weakening the variation exponent. The Brownian enhancement is geometric for every $p>2$.

For $2<p<3$ and a continuous path $h$ of bounded variation, translate the first level by $h$ and add the mixed integrals $\int X_{s,r}\otimes d{h}_r$, $\int h_{s,r}\otimes d{X}_r$ and $\int h_{s,r}\otimes d{h}_r$ to the second level. Integration by parts defines the second mixed integral. The operation is continuous in the rough path metric and the variation norm of $h$. For canonical lifts it is precisely the enhancement of the sum of the paths.

For a deterministic path $h$ in the Cameron-Martin space of Wiener measure, the law of $T_h\mathbf{B}$ is equivalent to the law of enhanced Brownian motion. The Cameron-Martin theorem gives equivalence for the underlying Brownian paths; the canonical enhancement of the translated path is exactly the rough-path translation, including its mixed area terms. Continuity of this translation transfers small-ball positivity at the identity to every control signature.

A weak geometric $p$-rough path is a continuous path in the free step-N nilpotent Lie group, with $N=\lfloor p\rfloor$, whose increments have finite $p$-variation for the Carnot-Carathéodory distance. Its multiplicative increments are ${\mathbf{X}}_{s,t}={\mathbf{X}}_s^{-1}\otimes {\mathbf{X}}_t$. They obey the Chen identity and the algebraic symmetries of path signatures.

The first level of this group path is zero but its second level grows linearly. Its Carnot-Carathéodory distance increments are a constant times $|t-s{|}^{1/2}$. It therefore has finite $p$-variation precisely when $p\ge 2$, and is a step-two weak geometric rough path at $p=2$, even though its first-level trace is constant. For higher truncations the same Lie exponential defines a compatible lift.