The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point
We consider a stochastic differential equation on a domain D in n-dimensional real space, where the associated dynamical system is linear, and D contains a repulsive type stationary point at the origin O. We obtain an exit law for the first exit time of the solution process from a ball of arbitrary...
Main Author: | |
---|---|
Other Authors: | |
Format: | Others |
Published: |
Virginia Tech
2014
|
Subjects: | |
Online Access: | http://hdl.handle.net/10919/28703 http://scholar.lib.vt.edu/theses/available/etd-08182003-173249/ |
id |
ndltd-VTETD-oai-vtechworks.lib.vt.edu-10919-28703 |
---|---|
record_format |
oai_dc |
spelling |
ndltd-VTETD-oai-vtechworks.lib.vt.edu-10919-287032020-09-26T05:34:33Z The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point Buterakos, Lewis Allen Mathematics Day, Martin V. Boisen, Monte B. Jr. Rossi, John F. Ball, Joseph A. Kohler, Werner E. repulsive stationary point dynamical systems asymptotics random perturbations stochastic differential equations We consider a stochastic differential equation on a domain D in n-dimensional real space, where the associated dynamical system is linear, and D contains a repulsive type stationary point at the origin O. We obtain an exit law for the first exit time of the solution process from a ball of arbitrary radius centered at the origin, which involves additive scaling as in Day (1995). The form of the scaling constant is worked out and shown to depend on the structure of the Jordan form of the linear drift. We then obtain an extension of this exit law to the first exit time of the solution process from the general domain D by considering the exit in two stages: first from the origin O to the boundary of the ball, for which the aforementioned exit law applies, and then from the boundary of the ball to the boundary of D. In this way we are able to determine for which Jordan forms we can obtain a limiting distribution for the first exit time to the boundary of D as the noise approaches 0. In particular, we observe there are cases for which the exit time distribution diverges as the noise approaches 0. Ph. D. 2014-03-14T20:15:22Z 2014-03-14T20:15:22Z 2003-08-04 2003-08-18 2003-08-22 2003-08-22 Dissertation etd-08182003-173249 http://hdl.handle.net/10919/28703 http://scholar.lib.vt.edu/theses/available/etd-08182003-173249/ dissertation.pdf In Copyright http://rightsstatements.org/vocab/InC/1.0/ application/pdf Virginia Tech |
collection |
NDLTD |
format |
Others
|
sources |
NDLTD |
topic |
repulsive stationary point dynamical systems asymptotics random perturbations stochastic differential equations |
spellingShingle |
repulsive stationary point dynamical systems asymptotics random perturbations stochastic differential equations Buterakos, Lewis Allen The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point |
description |
We consider a stochastic differential equation on a domain D in n-dimensional real space, where the associated dynamical system is linear, and D contains a repulsive type stationary point at the origin O. We obtain an exit law for the first exit time of the solution process from a ball of arbitrary radius centered at the origin, which involves additive scaling as in Day (1995). The form of the scaling constant is worked out and shown to depend on the structure of the Jordan form of the linear drift. We then obtain an extension of this exit law to the first exit time of the solution process from the general domain D by considering the exit in two stages: first from the origin O to the boundary of the ball, for which the aforementioned exit law applies, and then from the boundary of the ball to the boundary of D. In this way we are able to determine for which Jordan forms we can obtain a limiting distribution for the first exit time to the boundary of D as the noise approaches 0. In particular, we observe there are cases for which the exit time distribution diverges as the noise approaches 0. === Ph. D. |
author2 |
Mathematics |
author_facet |
Mathematics Buterakos, Lewis Allen |
author |
Buterakos, Lewis Allen |
author_sort |
Buterakos, Lewis Allen |
title |
The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point |
title_short |
The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point |
title_full |
The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point |
title_fullStr |
The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point |
title_full_unstemmed |
The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point |
title_sort |
exit time distribution for small random perturbations of dynamical systems with a repulsive type stationary point |
publisher |
Virginia Tech |
publishDate |
2014 |
url |
http://hdl.handle.net/10919/28703 http://scholar.lib.vt.edu/theses/available/etd-08182003-173249/ |
work_keys_str_mv |
AT buterakoslewisallen theexittimedistributionforsmallrandomperturbationsofdynamicalsystemswitharepulsivetypestationarypoint AT buterakoslewisallen exittimedistributionforsmallrandomperturbationsofdynamicalsystemswitharepulsivetypestationarypoint |
_version_ |
1719341545935601664 |