Second-order differential equations with random perturbations and small parameters

We consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small paramet...

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Bibliographic Details
Published inProceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 147; no. 4; pp. 763 - 779
Main Authors Kamenskii, M., Pergamenchtchikov, S., Quincampoix, M.
Format Journal Article
LanguageEnglish
Published Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.08.2017
Cambridge University Press
Royal Society of Edinburgh
Subjects
Asymptotic properties
Boundary value problems
Brownian motion
Differential equations
Existence theorems
Mathematical analysis
Mathematics
Probability
Probability theory
Randomness
Uniqueness theorems
34D15
34B05
34B27
boundary-value problems
60J60
Green functions
stochastic averaging method
Primary 60H10
58J37
34C29
Green functions AMS 2000 Subject Classifications: Primary: 60H10
Boundary value problems
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ISSN0308-2105
1473-7124
DOI10.1017/S0308210516000354

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Summary:We consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero.
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content type line 14
ISSN:0308-2105
1473-7124
DOI:10.1017/S0308210516000354