David Kelly, CIMS

Abstract:

It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic. One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the statistical properties of chaotic systems can be well approximated by stochastic differential equations.

In this talk, we focus on fast-slow ODEs, where the fast, chaotic variables are fed into the slow variables to yield a diffusion approximation. In particular we focus on the case where the fast process is multi-dimensional and enters the slow equation as multiplicative “noise”. The tools from rough path theory prove useful in this difficult setting. From a stochastic modeling perspective, the limiting slow variables are somewhat surprising. Even though the noise is approximated by a smooth (chaotic) signal, one does not obtain a Stratonovich integral in the limiting equation.