This thesis is concerned with studying homogenization properties of fast-slow sys-
tems driven by fractional noise. We consider various fast-slow systems of the form
dx εt = α(ε)f (x εt , y t ε )dt,
x ε 0 = x 0 ,
where α(ε) is our scaling, and the fast process y t ε injects the noise.
Due to the fractional nature of our noise, usual methods for Markov processes
or martingales are not available, whence we rely on the continuity theorem for
rough differential equations in order to reduce our problem to establishing limit
theorems in rough path spaces. This follows the recipe of first proving convergence
in finite-dimensional distributions and then tightness via suitable Kolmogorov-like
L p bounds on increments.
We focus our investigation on fast processes given by multiple stochastic in-
tegrals, which allows us to make use of the well-known fourth moment theorem
as well as L 2 kernel convergence. Depending on the regime, we obtain Wiener
processes, fractional Brownian motions or Hermite processes as limits. To obtain
the tightness estimates, we primarily rely on Gaussian methods such as the dia-
gram formula and hypercontractivity, which allow us, assuming a fast decay of the
coefficients in the relevant Wiener chaos expansion, to resort to L 2 bounds.
• In case y t ε is a rescaled fractional Ornstein-Uhlenbeck process and f is of
product form, i.e. f (x, y) = g(x)h(y), we obtain a multi-scale homogeniza-
tion result stating that x εt converges weakly in Hölder spaces towards the
solution of a stochastic differential equation driven by Wiener and/or Her-
mite processes.
• In case y t ε is a moving average of a kernel integrated against a Wiener process,
we obtain weak convergence in the supnorm topology of x εt to an Itô diffusion.
• In case y t ε is a moving average of a suitable kernel integrated against a
Hermite process and f is of product form, we obtain convergence of x εt to
a stochastic differential equation either driven by a Wiener or a Hermite
process.