Communications in Mathematical Sciences

Volume 13 (2015)

Number 2

Stochastic mode-reduction in models with conservative fast sub-systems

Pages: 297 – 314



Ankita Jain (Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana, U.S.A.)

Ilya Timofeyev (Department of Mathematics, University of Houston, Texas, U.S.A.)

Eric Vanden-Eijnden (Courant Institute, New York University, New York, N.Y., U.S.A.)


A stochastic mode reduction strategy is applied to multiscale models with a deterministic energy-conserving fast sub-system. Specifically, we consider situations where the slow variables are driven stochastically and interact with the fast sub-system in an energy-conserving fashion. Because the stochastic terms only affect the slow variables, the fast sub-system evolves deterministically on a sphere of constant energy. However, in the full model the radius of the sphere slowly changes due to the coupling between the slow and fast dynamics. Therefore, the energy of the fast sub-system becomes an additional hidden slow variable that must be accounted for in order to apply the stochastic mode reduction technique to systems of this type.


mode-reduction, conservative fast sub-system

2010 Mathematics Subject Classification

60J60, 62Mxx

Published 3 December 2014