Rough Paths

Job: give a pathwise calculus for controlled ODEs driven by irregular signals (beyond Young / Itô), via signatures and rough path lifts — analytic backbone linking to regularity structures.

Core

Classical ODE \(dy=f(y)\,dx\) needs \(x\) of bounded variation or Young (\(p\)-variation \(p<2\)). Brownian paths have \(p\)-variation for all \(p>2\) — Itô/Stratonovich use probability. Rough path theory (Lyons) rebuilds the integral pathwise from enhanced data \((X,\mathbb{X})\) (path + iterated integrals).

Layer Content
Controlled ODEs Riemann–Stieltjes / Young regime
\(p\)-variation roughness scale; Young \(p^{-1}+q^{-1}>1\)
Signature tensor series of iterated integrals
Lyons rough paths multiplicative functionals; Chen
Probability link Itô / Stratonovich BM as rough paths
SPDE link regularity structures as branched upgrade

Reading map

Page Focus
Foundations controlled ODEs, \(p\)-variation, signature, Lyons, Itô vs rough, regularity structures pointer
Signature & Features Chen, Hambly–Lyons, truncated/\(G^n\), expected signature, ML kernel
Controlled & Sewing Gubinelli controlled, sewing lemma, rough integral, Itô–Lyons map
FBM & Non-Semimartingale fBM \(H>1/4\) canonical lift, non-semimartingale, RDE
Branched & Hopf branched trees, Connes–Kreimer Hopf, regularity structures upgrade

Dependency order

  1. Multivariable calculus / ODE fundamentals.
  2. Stochastic processes (BM, Itô) for probabilistic examples.
  3. \(p\)-variation and Young \(\to\) rough paths \(\to\) RDE existence.
  4. Optional: SPDE and regularity structures.

Interfaces

Area Contact
Stochastic processes BM, semimartingales as rough paths
Martingales quadratic variation \(=\) second level
SPDE Hairer regularity structures
Functional analysis Banach-valued paths; sewing lemma
Geometric measure theory variation, rectifiability (cousin metrics)

Notation (paper-standard)

  • \(X\in C^{p\text{-var}}([0,T],V)\); \(\mathbb{X}_{s,t}\in V\otimes V\)
  • \(\mathbf{X}=(X,\mathbb{X})\) rough path; \(\delta\mathbb{X}=\Delta X\otimes\Delta X\) Chen
  • \(\mathscr{S}(X)\) signature; \(\mathcal{C}^\alpha\) Hölder rough paths \(\alpha\in(1/3,1/2]\)

See it

Brownian motion as a rough path. Enhance \(B_t\) by \(\mathbb{B}_{s,t}=\int_s^t (B_u-B_s)\otimes dB_u\) in Itô or Stratonovich sense. The pair \((B,\mathbb{B})\) is a.s.\ a geometric (Stratonovich) or non-geometric (Itô) rough path of Hölder roughness \(<1/2\). RDE \(dY=f(Y)\,d\mathbf{B}\) recovers Stratonovich SDE when \(\mathbf{B}\) is geometric — pathwise continuous dependence on the driving rough path (Lyons continuity theorem) is the stability statement Itô calculus lacks in uniform topology on raw paths.

Check

Why does Young integration fail for Brownian motion?

BM has finite \(p\)-variation a.s.\ iff \(p>2\). Young needs \(1/p+1/q>1\) for integrator/integrand variations; the natural integrand \(f(Y)\) tracks the same roughness as \(B\), so \(p=q>2\) gives \(1/p+1/q<1\) — outside Young’s range.

What extra data does a rough path add beyond the path \(X\)?

A second-level (and higher, if \(p\ge 3\)) process \(\mathbb{X}_{s,t}\) playing the role of \(\int_s^t X_{s,u}\otimes dX_u\), subject to Chen’s relation and analytic estimates \(|\mathbb{X}_{s,t}|=O(|t-s|^{2/p})\).

Pitfalls

  • Geometric vs non-geometric: geometric rough paths are limits of smooth paths in rough metric; Stratonovich lifts are geometric, Itô are not (need branched / weak geometric variants).
  • \(p\)-variation vs Hölder: related but not identical topologies; both used in the literature.
  • Signature uniqueness: tree-like equivalence — signature determines path only modulo tree-like excursions.

Refs

  • Lyons, Differential equations driven by rough signals (Rev.\ Mat.\ Iberoamericana)
  • Lyons–Caruana–Lévy, Differential Equations Driven by Rough Paths (LNM)
  • Friz–Hairer, A Course on Rough Paths
  • Friz–Victoir, Multidimensional Stochastic Processes as Rough Paths

Next

Foundations · Stochastic processes · Martingales · SPDE · PDE · Signature & Features · Controlled & Sewing · FBM & Non-Semimartingale · Branched & Hopf