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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 which are discussed in detail. © 2019 The Author(s
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
which are discussed in detail.Comment: Final version to appear in Journal of Functional Analysi
G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion
The present paper is devoted to the study of sample paths of G-Brownian
motion and stochastic differential equations (SDEs) driven by G-Brownian motion
from the view of rough path theory. As the starting point, we show that
quasi-surely, sample paths of G-Brownian motion can be enhanced to the second
level in a canonical way so that they become geometric rough paths of roughness
2 < p < 3. This result enables us to introduce the notion of rough differential
equations (RDEs) driven by G-Brownian motion in the pathwise sense under the
general framework of rough paths. Next we establish the fundamental relation
between SDEs and RDEs driven by G-Brownian motion. As an application, we
introduce the notion of SDEs on a differentiable manifold driven by GBrownian
motion and construct solutions from the RDE point of view by using pathwise
localization technique. This is the starting point of introducing G-Brownian
motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin.
The last part of this paper is devoted to such construction for a wide and
interesting class of G-functions whose invariant group is the orthogonal group.
We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian
motion of independent interest
Rough Burgers-like equations with multiplicative noise
We construct solutions to vector valued Burgers type equations perturbed by a multiplicative space–time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise