- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- BIRS Workshop Lecture Videos /
- Small noise limit for singularly perturbed diffusion
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
Small noise limit for singularly perturbed diffusion Athreya, Siva
Description
We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by \begin{eqnarray*} X^{\varepsilon}_t &=& x_0 + \int_0^t b(X^{\varepsilon}_s, Y^{\varepsilon}_s)ds + \varepsilon^{\alpha}B_t, \label{ex} \\ Y^{\varepsilon}_t &=& y_0 - \frac{1}{\varepsilon} \int_0^t \nabla_yU(X^{\varepsilon}_s, Y^{\varepsilon}_s)ds+ \frac{s(\varepsilon)}{\sqrt{\varepsilon}} W_t,\label{wye} \end{eqnarray*} where $x_0 \in {\mathbb R}^d, y_0 \in {\mathbb R}^m$, $B_t, W_t$ are independent Brownian motions, $b : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}^d$, $U : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}$, and $s : (0, \infty) \rightarrow (0, \infty).$ One observes that there is a time scale separation between $X$ and $Y$. Under suitable assumptions on $b, U$, for $0 < \alpha < \frac{1}{2}$, if $s(\epsilon) \rightarrow 0$ goes to zero at a prescribed slow enough rate then we establish all weak limits points of $X^{\epsilon}$, as $\epsilon \rightarrow 0$, as Fillipov solutions to a differential inclusion. This is joint work with V. Borkar, S. Kumar and R. Sundaresan.
Item Metadata
Title |
Small noise limit for singularly perturbed diffusion
|
Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
|
Date Issued |
2017-10-24T17:13
|
Description |
We consider a simultaneous small noise limit for a
singularly perturbed coupled diffusion
described by
\begin{eqnarray*}
X^{\varepsilon}_t &=& x_0 + \int_0^t b(X^{\varepsilon}_s, Y^{\varepsilon}_s)ds + \varepsilon^{\alpha}B_t,
\label{ex} \\
Y^{\varepsilon}_t &=& y_0 - \frac{1}{\varepsilon} \int_0^t \nabla_yU(X^{\varepsilon}_s, Y^{\varepsilon}_s)ds+
\frac{s(\varepsilon)}{\sqrt{\varepsilon}} W_t,\label{wye}
\end{eqnarray*}
where $x_0 \in {\mathbb R}^d, y_0 \in {\mathbb R}^m$, $B_t, W_t$ are independent Brownian motions, $b : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}^d$, $U : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}$, and $s : (0, \infty) \rightarrow (0, \infty).$
One observes that there is a time scale separation between $X$ and $Y$. Under suitable assumptions on $b, U$, for $0 < \alpha < \frac{1}{2}$, if $s(\epsilon) \rightarrow 0$ goes to zero at a prescribed slow enough rate then we establish all weak limits points of $X^{\epsilon}$, as $\epsilon \rightarrow 0$, as Fillipov solutions to a differential inclusion.
This is joint work with V. Borkar, S. Kumar and R. Sundaresan.
|
Extent |
45 minutes
|
Subject | |
Type | |
File Format |
video/mp4
|
Language |
eng
|
Notes |
Author affiliation: Indian Statistical Institute Bangalore Centre
|
Series | |
Date Available |
2018-04-23
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0365955
|
URI | |
Affiliation | |
Peer Review Status |
Unreviewed
|
Scholarly Level |
Faculty
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International