We investigate the two-point stochastic boundary-value problem on [0, 1]: whereẆ is a white noise on [0, 1], ξ and η 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. ξ and η, (0) has a unique solution a.s. However, as soon as f is non-linear, bifurcation appears. We

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... stigate 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.