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2019 | OriginalPaper | Buchkapitel

Multiscale Systems, Homogenization, and Rough Paths

verfasst von : Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne, Huilin Zhang

Erschienen in: Probability and Analysis in Interacting Physical Systems

Verlag: Springer International Publishing

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Abstract

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479–520], taking into account recent progress in p-variation and càdlàg rough path theory.

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Vorheriges Kapitel Yang–Mills for Probabilists
Nächstes Kapitel The Deterministic and Stochastic Shallow Lake Problem
Fußnoten
1
Since our limit processes here—a Brownian motion—is continuous, there is no need to work with the Skorokhod topology on D.
 
2
In view of the genuine non-linearity of rough path spaces, we refrain from writing \(\Vert \mathbf {X}- \tilde{\mathbf {X}} \Vert _{p\text {-var},[0,1]}\).
 
3
In coordinates, when \(\mathcal {B}= \mathbb {R}^m\), we have \(DV (Y_s) V (Y_s) \mathbb {X}_{s,t} = \partial _\alpha V_\gamma (Y_s) V^\alpha _\beta (Y_s) \mathbb {X}_{s,t}^{\beta ,\gamma }\) with summation over \(\alpha = 1, \ldots , d\) and \(\beta , \gamma = 1, \ldots , m.\).
 
4
Often \(B^n\) has continuous BV sample paths. Every such process is (trivially) a semimartingale (under its own filtration); the Stratonovich SDE interpretation is the one consistent with the ODE interpretation, in the sense of a Riemann-Stieltjes integral equation.
 
5
Again it suffices to work with the uniform topology on both \({\pmb {\mathscr {C}}}\) and \({\pmb {\mathscr {D}}}\).
 
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Metadaten
Titel
Multiscale Systems, Homogenization, and Rough Paths
verfasst von
Ilya Chevyrev
Peter K. Friz
Alexey Korepanov
Ian Melbourne
Huilin Zhang
Copyright-Jahr
2019
Buch
Probability and Analysis in Interacting Physical Systems

Print ISBN: 978-3-030-15337-3

Electronic ISBN: 978-3-030-15338-0

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-15338-0_2

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