Branched Rough Paths, the Hopf Algebra, and the Regularity Structures¶
Job: develop the branched rough paths (Gubinelli trees), the Connes–Kreimer Hopf algebra, the regularity structures upgrade, and the renormalization — with the branched / Connes–Kreimer / regularity structures as the worked example.
Core¶
The branched rough paths (Gubinelli)¶
| Branched / trees / Hopf / tensor | Description |
|---|---|
| Branched rough paths | the trees (the noncommutative, the Hopf, the beyond the tensor) |
| Trees | the rooted trees the iterated integrals (the branched, the non-tensor) |
| Hopf / tensor | the branched the Hopf, the tensor the symmetric (the geometric the tensor) |
The Hopf algebra and the Connes–Kreimer¶
| Hopf / Connes–Kreimer / NCG | Description |
|---|---|
| Connes–Kreimer | the Hopf algebra of rooted trees (the renormalization, the NCG) |
| Hopf algebra | the coproduct \(\Delta\), the product, the antipode (the Hopf, the NCG) |
The regularity structures and the branched¶
| Regularity structures / branched / SPDE | Description |
|---|---|
| Hairer regularity structures | the SPDE / regularity structures, the branched the upgrade |
| Branched / regularity | the branched the regularity structures (the SPDE, the upgrade) |
The renormalization and the branched¶
| Renormalization / BPHZ / branched / counterterms | Description |
|---|---|
| Renormalization | the branched the renormalization (the BPHZ, the SPDE / regularity structures) |
| BPHZ | the Bogoliubov–Parasiuk–Hepp–Zimmermann (the counterterms, the SPDE / regularity structures) |
See it¶
Branched rough paths and the Connes–Kreimer Hopf algebra / the regularity structures. The branched: the trees (the noncommutative, the Hopf, the beyond the tensor). The Connes–Kreimer: the Hopf algebra of trees (the renormalization, the NCG). The regularity structures: the Hairer (the SPDE / regularity structures, the branched the upgrade, the SPDE). The key point: the branched the Connes–Kreimer / the regularity structures — the rough path the Hopf / the SPDE.
Check¶
Explain the branched rough paths and the Hopf algebra of trees, and the Connes–Kreimer.
The branched: the trees the iterated integrals (the noncommutative, the Hopf, the beyond the tensor — the geometric the tensor / the symmetric, the branched the trees / the noncommutative). The Connes–Kreimer: the Hopf algebra of rooted trees (the coproduct \(\Delta(\tau)=\sum\) the admissible cuts, the product the grafting, the antipode the renormalization, the NCG). The branched / the Connes–Kreimer: the branched the Connes–Kreimer the Hopf (the trees the Hopf, the rough path the branched). So the branched the trees / the Hopf, the Connes–Kreimer the renormalization.
Explain the regularity structures as the branched upgrade, and the BPHZ renormalization.
The regularity structures (Hairer): the SPDE / regularity structures, the model / the modelled distributions / the reconstruction (the SPDE). The branched the upgrade: the branched rough paths the 1D / the paths, the regularity structures the multi-D / the distributions (the branched the intermediate, the regularity the full). The BPHZ: the renormalization of the branched / the regularity (the counterterms, the SPDE / regularity structures, the BPHZ the Bogoliubov–Parasiuk–Hepp–Zimmermann). So the regularity the branched upgrade, the BPHZ the renormalization.
Pitfalls¶
- The branched the trees (the noncommutative, the Hopf, the beyond the tensor) — the branched the trees, the geometric the tensor.
- The Connes–Kreimer the Hopf (the trees the Hopf, the renormalization, the NCG) — the Connes–Kreimer the Hopf, the NCG.
- The regularity structures the upgrade (the SPDE / regularity structures, the branched the intermediate) — the regularity the full, the branched the 1D.
- The BPHZ the renormalization (the counterterms, the SPDE / regularity structures) — the BPHZ the renormalized, the SPDE the singular.
Refs¶
- Gubinelli, "Ramified rough paths" (2013); the branched rough paths
- Connes–Kreimer, "Hopf algebras, renormalization and noncommutative geometry" (1998–2000); the Hopf
- Hairer, "A theory of regularity structures" (Invent. 2014); the SPDE / regularity structures
- Chandra–Hairer, An Introduction to Regularity Structures (2018); the branched / the regularity
Next¶
RP index · Foundations · Signature & Features · Controlled & Sewing · FBM & Non-Semimartingale · SPDE / regularity structures · NCG