Functional limit theorems for volterra processes and applications to
homogenization
homogenization
File(s)
Accepted version
Author(s)
Gehringer, Johann
Li, Xue-Mei
Sieber, Julian
Type
Working Paper
Abstract
We prove an enhanced limit theorem for additive functionals of a
multidimensional Volterra process $(y_t)_{t\geq 0}$. As an application, we
establish weak convergence of the solutions of rough differential equations
(RDE) of the form $$
dx^\varepsilon_t=\frac 1 {\sqrt \varepsilon}
f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})\,dt+g(x_t^\varepsilon)\,d\mathbf{B}_t,$$
and identify their limits as solutions of an RDE driven by a Gaussian field
with a drift coming from the L\'evy area correction of the limiting rough
driver. The equation models a passive tracer in a random field.
In particular if $h$ is random field such that $h(x, \cdot)$ a
semi-martingale with spatial parameter $x$, we show that the solutions of the
equations $$ dx^\varepsilon_t=\frac 1 {\sqrt \epsilon}
f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})\,dt+h(x_t^\varepsilon, dt),$$
converge weakly to that of a Kunita type It\^o SDE $dx_t=G(x_t,dt)$ where
$G(x,t)$ is a semi-martingale with spatial parameters. Furthermore the
$N$-point motions converge.
multidimensional Volterra process $(y_t)_{t\geq 0}$. As an application, we
establish weak convergence of the solutions of rough differential equations
(RDE) of the form $$
dx^\varepsilon_t=\frac 1 {\sqrt \varepsilon}
f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})\,dt+g(x_t^\varepsilon)\,d\mathbf{B}_t,$$
and identify their limits as solutions of an RDE driven by a Gaussian field
with a drift coming from the L\'evy area correction of the limiting rough
driver. The equation models a passive tracer in a random field.
In particular if $h$ is random field such that $h(x, \cdot)$ a
semi-martingale with spatial parameter $x$, we show that the solutions of the
equations $$ dx^\varepsilon_t=\frac 1 {\sqrt \epsilon}
f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})\,dt+h(x_t^\varepsilon, dt),$$
converge weakly to that of a Kunita type It\^o SDE $dx_t=G(x_t,dt)$ where
$G(x,t)$ is a semi-martingale with spatial parameters. Furthermore the
$N$-point motions converge.
Date Issued
2021-04-13
Citation
2021
Publisher
arXiv
Copyright Statement
© 2021 The Author(s)
Identifier
http://arxiv.org/abs/2104.06364v1
Subjects
math.PR
math.PR
34F05, 60F05, 60F17
Notes
30 pages
Publication Status
Published