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Zbl 0566.60058
Deimling, K.; Ladde, G.S.; Lakshmikantham, V.
Sample solutions of stochastic boundary value problems.
(English)
[J] Stochastic Anal. Appl. 3, 153-162 (1985). ISSN 0736-2994; ISSN 1532-9356/e

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). \par 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.
MSC 2000:
*60H10 Stochastic ordinary differential equations
60H25 Random operators, etc.
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
34B15 Nonlinear boundary value problems of ODE

Keywords: existence theorems; stochastic Sturm-Liouville problems; extensions of classical comparison techniques; measurable selections

Citations: Zbl 0555.60036

Cited in: Zbl 0951.47047 Zbl 0591.60051

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