Convergence of fast-slow ODEs to stochastic differential
equations
April 7, 2pm; 646 PGH
Abstract
A project started recently with Andrew Stuart investigates the convergence of
certain deterministic systems to a stochastic differential equation.
As a starting point, we consider the (over-simplistic) fast-slow system
x' = ε-1f0(y) + f(x,y)
y' = ε-2g(y)
where the fast y-variables lie in an attractor with invariant
measure μ and the slow x-variables lie
in Rd. It is assumed that ∫ f0
dμ=0.
We prove that under very mild conditions on the fast variables, solutions
xε converge weakly
in C([0,T],Rd) as ε→0 to
solutions X of the stochastic differential equation
X'= W' + F(X),
where W is a d-dimensional Brownian motion and F(x)=∫
f(x,y) dμ(y).
A major difference between our approach and related projects is that we do
not rely on decay of correlations for the y' equation (decay of
correlations for flows is a notoriously difficult and poorly understood
problem). Instead we use invariance principles (a generalisation of the
central limit theorem giving convergence to Brownian motion) and large
deviation estimates which have been derived for a very large class of
systems in collaboration with Matthew Nicol.
This talk will include an introduction to the ideas needed from probability
theory.
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Last modified: April 08 2016 - 20:30:35
