Ramification of rough paths

By Massimiliano Gubinelli. 2006

209 citations recorded by the source corpus.

The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen's multiplicative property involves the Durr–Connes–Kreimer coproduct on rooted trees. This turns out to be the natural setting for a non-geometric theory of rough paths.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/10f51082291a3ea7314516d54c94feb14a64d2bb

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Differential equations driven by rough paths with jumps (2017, methodology): “For example, as pointed out in the works of Gubinelli and Hairer–Kelly, [18, 20], the 2In absence of jumps forward and geometric solutions coincide.”
A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion (2010, background): “up to order n = 1/γ can be defined, see [21].”
Lectures on singular stochastic PDEs (2015, background): “In [12, 13], the notion of controlled paths has been introduced in order to extend the applicability of the rough path ideas to a larger class of problems that are not necessarily related to the integration of ODEs but which still retain the one–dimensional nature of the directions in which the irre”
A theory of regularity structures (2013, background): “This is because the regularity structure T (r) F built in this section does not encode the Leibniz rule, so that it can accommodate the type of effects described in [55,61,62] (or even just Itô’s formula in one dimension) which cannot arise when only considering lifts of smooth functions.”
Non-linear rough heat equations (2009, background): “This particular case has been dealt with in [25], as well as the polynomial case, for which specific and individual treatments based on trees-indexed integral [21,23] are suggested.”
A regularity structure for rough volatility (2017, methodology): “At this stage, a tree notation (omnipresent in the works of Hairer) would come in handy and we refer to Bruned, Chevyrev, Friz, amd Preiß (2019) (and the references therein) for a recent attempt to reconcile the tree formalism of branched rough path (Gubinelli, 2010; Hairer & Kelly, 2015) and the mo”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, background): “Limits can then be taken thanks to [19], which completes the proof.”