BPHZ Renormalization in Gaussian Rough Paths

A renormalized rough path over fractional Brownian motion

We construct in this article a rough path over fractional Brownian motion with arbitrary Hurst index by (i) using the Fourier normal ordering algorithm introduced in \cite{Unt-Holder} to reduce the problem to that of regularizing tree iterated integrals and (ii) applying the Bogolioubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization algorithm to Feynman diagrams representing tree iterated integrals.

An analytic BPHZ theorem for regularity structures

We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated framework for studying these problems but previous works had to expend significant effort to obtain these stochastic estimates in an ad-hoc manner. In contrast, the main result of this article operates as a black box which automatically produces these estimates for nearly all of the equations that fit within the scope of the theory of regularity structures. Our approach leverages multi-scale analysis s…

从非几何到几何粗糙路径的重整化

Renormalisation from non-geometric to geometric rough paths

本文比较了基于Hairer-Kelly映射和Lyons-Victoir定理的新粗糙路径重整化方法与BPHZ等现有方法，证明了其可交换性并给出显式公式，同时考察了替代方法中的重整化行为。

关于正则结构中的方差重整化——二维gPAM的情况

Variance renormalisation in regularity structures -- the case of $2d$ gPAM

本文研究了二维gPAM中因噪声粗糙性导致的方差重整化问题，通过提升噪声模型并结合BPHZ方法实现收敛。

从 Stratonovich 到 Itô 的分数布朗运动粗糙路径重整化

Rough Path Renormalization from Stratonovich to Itô for Fractional Brownian Motion

本文基于 Lyons 粗糙路径框架，为 Hurst 参数在 (1/3, 1/2] 区间的分数布朗运动建立 Itô 型路径积分理论，并将其应用于分数 Black-Scholes 模型和高维分数 Ornstein-Uhlenbeck 过程，解决了非半鞅过程的积分难题。

A Rough Path Perspective on Renormalization

We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a regularity structure in the sense of Hairer. Pre-Lie structures are seen to play a fundamental rule which allow a direct understanding of the translated (i.e. renormalized) equation under consideration. This construction is also novel with regard to the algebraic renormalization theory for regularity structures due to Bruned--Hairer--Zambotti (2016), the links with w…