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Sample solutions of stochastic boundary value problems. (English) Zbl 0566.60058
We prove existence theorems for stochastic Sturm-Liouville problems in case the nonlinearity f(t, x, x’, $\omega)$ is continuous in (x, x’) and measurable in (t, $\omega)$. Solutions are functions x:[0,1]$\times \Omega \to {\bbfR}$ such that x(t,$\cdot)$ is measurable and x’($\cdot,\omega)$ is a.c. We have no restrictions concerning the growth of f w.r. to $\omega$. Since f is only measurable in t, we need unusual extensions of classical comparison techniques. Since for fixed $\omega$ we do not have uniqueness of solutions, we need new results about measurable selections of fixed points and some related tricks used before by the first author, ibid. 3, 15-21 (1985; Zbl 0555.60036). We allow quadratic growth in x’ (Nagumo condition), but then we have to assume that f is bounded in t. It is not clear whether this restriction can be removed. Since the basic existence problems have been solved in this paper our suggestion for further studies is to establish $L\sp 2$- properties of the solution processes under reasonable assumptions about f and the boundary conditions, a problem as difficult as the ones solved here.

60H10Stochastic ordinary differential equations
60H25Random operators and equations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34B15Nonlinear boundary value problems for ODE
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