12 papers
Random dynamical systems generated by rough differential equations driven by controlled rough paths
Xing Gao, Jiaqi Liu
We study rough differential equations driven by controlled rough paths with drift. By means of the canonical lift of the controlled driver, we establish pathwise well-posedness and…
The planar Hopf algebra of noncommutative multi-indices
Yong Yu, Xing Gao, Dominique Manchon
We construct the planar Linares--Otto--Tempelmayr Hopf algebra, thereby filling the missing planar noncommutative multi-index corner in the square relating the LOT, Butcher--Connes…
Exact Signature Tail Asymptotics for Pure Rough Paths
Nannan Li, Xing Gao
We prove~\cite[Conjecture 2.12]{BGS20} on the signature tail asymptotics of pure rough paths and extend it to arbitrary reasonable tensor norms. In more details, let \[ \mathbf X_t…
A Low-Regularity Semigroup Sewing Lemma via Quotient Structures
Nannan Li, Xing Gao
We develop a low-regularity Sewing theory for the semigroup coboundary associated with a strongly continuous semigroup . Unlike the ordinary low-regularity Sewing pr…
Jump Itô-type formula with arbitrary regularity
Nannan Li, Xing Gao
We establish an Itô-type formula for finite -variation paths with jumps for arbitrary . The formula is stated in a fully pathwise form and separates the reduced rough i…
Rough differential equations and reduced rough paths: a Lie bracket characterization
Nannan Li, Xing Gao
This paper studies rough differential equations from the viewpoint of reduced rough paths in the Hölder regime \(\frac13<α\le\frac12\). A reduced rough path retains the first level…
Itô formula for reduced rough paths
Nannan Li, Xing Gao
The Itô formula, also known as the change-of-variables formula, is a cornerstone of Itô stochastic calculus. Over time, this formula has been extended to apply to random processes…
Controlled rough paths: a general Hopf-algebraic setting
Zhicheng Zhu, Xing Gao, Nannan Li +1
We set up controlled rough paths for a class of combinatorial Hopf algebras, encompassing shuffle, Butcher-Connes-Kreimer and Munthe-Kaas--Wright Hopf algebras. The class of contro…
Rough Burger-like SPDEs
Nannan Li, Xing Gao
We study a class of nonlinear Burgers-type stochastic partial differential equations driven by additive space-time white noise in one spatial dimension. Building on the rough path…
Itô formula for planarly branched rough paths
Nannan Li, Xing Gao
The Itô formula, originated by K. Itô, is focus on the stochastic calculus, where many stochastic processes can be placed under the framework of rough paths. In rough path theory,…
Rough differential equations and planarly branched universal limit theorem
Xing Gao, Nannan Li, Dominique Manchon
The universal limit theorem is a central result in rough path theory, which has been proved for: (i) rough paths with roughness ; (ii) geometric rou…
Free Novikov algebras and the Hopf algebra of decorated multi-indices
Zhicheng Zhu, Xing Gao, Dominique Manchon
We propose a combinatorial formula for the coproduct in a Hopf algebra of decorated multi-indices that recently appeared in the literature, which can be briefly described as the gr…