Peter Friz, Technische Universität Berlin

Multiscale Systems, Homogenization and Rough Paths

Rough paths (and its recent generalisations: paracontrolled calculus, regularity structures) provide a powerful framework to the analysis of (partial) differential equations equations. Typical applications include highly-oscillatory systems (a priori well-posed, but with unclear limiting behaviour) and stochastic equations (analytically ill-posed, though sometimes within reach of Ito's stochastic analysis). After presenting some general ideas, I will explain how rough paths have been used to solve a concrete fast-slow homogenisation problem originally posed in [5].

 

References:
[1] Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne, Huilin Zhang; Deterministic homogenization for discrete-time fast-slow systems under optimal moment assumptions. arXiv 2019
[2] Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne, Huilin Zhang; Multiscale systems, homogenization, and rough paths; arXiv 2017 and 2019 Springer Volume Varadhan 75
[3] Peter Friz, Paul Gassiat and Terry Lyons, Physical Brownian motion in a magnetic field as a rough path. Trans. AMS 2015
[4] David Kelly, Ian Melbourne, Deterministic homogenization for fast-slow systems with chaotic noise, Journal of Functional Analysis 2017
[5] Melbourne and A. M. Stuart; Diffusion limits of chaotic skew-product flows, Nonlinearity 2011.

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