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A Stochastic Two-Point Boundary Value Problem

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	A Stochastic Two-Point Boundary Value Problem
2.	Creator	Author's name, affiliation, country	S. J. Luo; FinancialCAD Corp.
2.	Creator	Author's name, affiliation, country	John B. Walsh; University of British Columbia
3.	Subject	Discipline(s)	Mathematics
3.	Subject	Keyword(s)	Stochastic boundary-value problems, bifurcations
3.	Subject	Subject classification	Primary. 60H10; Secondary. 34F05
4.	Description	Abstract	We investigate the two-point stochastic boundary-value problem on $[0,1]$: \begin{equation}\label{1} \begin{split} U'' &= f(U)\dot W + g(U,U')\\ U(0) &= \xi\\ U(1)&= \eta. \end{split} \tag{1} \end{equation} where $\dot W$ is a white noise on $[0,1]$, $\xi$ and $\eta$ are random variables, and $f$ and $g$ are continuous real-valued functions. This is the stochastic analogue of the deterministic two point boundary-value problem, which is a classical example of bifurcation. We find that if $f$ and $g$ are affine, there is no bifurcation: for any r.v. $\xi$ and $\eta$, (1) has a unique solution a.s. However, as soon as $f$ is non-linear, bifurcation appears. We investigate the question of when there is either no solution whatsoever, a unique solution, or multiple solutions. We give examples to show that all these possibilities can arise. While our results involve conditions on $f$ and $g$, we conjecture that the only case in which there is no bifurcation is when $f$ is affine.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2001-09-14
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/111
10.	Identifier	Digital Object Identifier	10.1214/EJP.v7-111
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 7
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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