A limit theorem for stochastic two-point boundary-value problems of ordinary differential equations

✍ Scribed by B. S. White; J. N. Franklin

• Publisher: John Wiley and Sons
• Year: 1979
• Tongue: English
• Weight: 624 KB
• Volume: 32
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160320203

✦ Synopsis

In this paper, we prove a limit theorem applicable to the asymptotic solution of stochastic two-point boundary-value and eigenvalue problems of a wide class of non-linear ordinary differential equations. Let E, 0 < E << 1, be a small parameter, and y ( s ) for 0 5 s <m a stochastic process. Then ~( x I E ) is a process which is rapidly varying on the x-space scale. We assume that the randomness enters the equation in this way, that is, only on the "fast" space scale X/E.

Consider first the prototype scalar equation, with random function f(X/E, 2): d2

With some probability, solutions of (1.1) may fail to exist, or may not be unique. We consider, alternatively, solutions of the related initial value problem with "shooting" parameter a :

For sufficiently smooth f , there will be a unique solution of (1.2) for each a. Solutions of (1.1) will then be obtained for random initial values 4' of (1.2) satisfying the non-linear stochastic boundary condition

(1.3) Ze(L, h e ) = 1 .