Abstract
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 language | American English |
|---|---|
| Pages (from-to) | 1305-1327 |
| Number of pages | 23 |
| Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |
| Volume | 58 |
| Issue number | 3 |
| DOIs | |
| State | Published - Aug 2022 |
Bibliographical note
Funding Information:The research of all three authors was supported in part by a European Advanced Grant (ERC AdG 320977 StochExtHo-mog) at the University of Warwick. The research of AK was supported in part by an Engineering and Physical Sciences Research Council grant (EP/P034489/1) at the University of Exeter.
Funding Information:
The research of all three authors was supported in part by a European Advanced Grant (ERC AdG 320977 StochExtHomog) at the University of Warwick. The research of AK was supported in part by an Engineering and Physical Sciences Research Council grant (EP/P034489/1) at the University of Exeter.
Publisher Copyright:
© Association des Publications de l'Institut Henri Poincaré, 2022.
Keywords
- Deterministic homogenization
- Fast-slow systems
- Iterated moment estimates
- Martingale decompositions
- Nonuniformly expanding maps and flows