Multiscale systems, homogenization, and rough paths

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.

arXiv comment27 pages. Minor corrections. To appear in Proceedings of the Conference in Honor of the 75th Birthday of S.R.S. Varadhan

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• Smooth approximation of stochastic differential equations
• A Note on Diffusion Limits of Chaotic Skew Product Flows
• Homogenization for Deterministic Maps and Multiplicative Noise
• Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps
• Canonical RDEs and general semimartingales as rough paths
• An invariance principle for maps with polynomial decay of correlations
• Random walks and Lévy processes as rough paths
• Superpolynomial and polynomial mixing for semiflows and flows
• Rough path recursions and diffusion approximations

Cited by in corpus (10)

• Superdiffusive limits for deterministic fast-slow dynamical systems
• Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion
• Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 2
• Rough semimartingales and $p$-variation estimates for martingale transforms
• Generating diffusions with fractional Brownian motion
• Variational Principles on Geometric Rough Paths and the Lévy Area Correction
• Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-mixing Nonuniformly Hyperbolic Maps
• Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas
• Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering
• Superdiffusive limits beyond the Marcus regime for deterministic fast-slow systems