Electronic Journal of Probability

A Stochastic Two-Point Boundary Value Problem

S. Luo and John Walsh

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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.

Article information

Electron. J. Probab., Volume 7 (2002), paper no. 12, 32 pp.

Accepted: 14 September 2001
First available in Project Euclid: 16 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

Stochastic boundary-value problems bifurcations

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Luo, S.; Walsh, John. A Stochastic Two-Point Boundary Value Problem. Electron. J. Probab. 7 (2002), paper no. 12, 32 pp. doi:10.1214/EJP.v7-111. https://projecteuclid.org/euclid.ejp/1463434885

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