Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 1

Alexey Korepanov, Zemer Kosloff, Ian Melbourne

Research output: Contribution to journalArticlepeer-review


We consider deterministic homogenization (convergence to a stochastic differential equation) for multiscale systems of the form xk+1 = xk + n-1an(xk,yk) + n-1/2bn(xk,yk), yk+1 = Tnyk, where the fast dynamics is given by a family Tn of nonuniformly expanding maps. Part 1 builds on our recent work on martingale approximations for families of nonuniformly expanding maps. We prove an iterated weak invariance principle and establish optimal iterated moment bounds for such maps. (The iterated moment bounds are new even for a fixed nonuniformly expanding map T .) The homogenization results are a consequence of this together with parallel developments on rough path theory in Part 2 by Chevyrev, Friz, Korepanov, Melbourne and Zhang.

Original languageAmerican English
Pages (from-to)1305-1327
Number of pages23
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number3
StatePublished - Aug 2022

Bibliographical note

Publisher Copyright:
© Association des Publications de l'Institut Henri Poincaré, 2022.


  • Deterministic homogenization
  • Fast-slow systems
  • Iterated moment estimates
  • Martingale decompositions
  • Nonuniformly expanding maps and flows


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