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Second-order differential equations with random perturbations and small parameters

Part of: Partial differential equations on manifolds; differential operators Qualitative theory Markov processes Stability theory Stochastic analysis Boundary value problems

Published online by Cambridge University Press:  05 July 2017

M. Kamenskii ,
S. Pergamenchtchikov  and
M. Quincampoix
M. Kamenskii
Affiliation:
Voronezh State University, Universitetskay pl. 1, 394063 Voronezh, Russia ([email protected])
S. Pergamenchtchikov
Affiliation:
Laboratoire de Mathématiques Raphael Salem, CNRS–UMR 6085, Université de Rouen, Avenue de l'Université, BP.12, Technopôle du Madrillet F76801, Saint-Étienne-du-Rouvray, France ([email protected]) and International Laboratory of Statistics of Stochastic Processes and Quantitative Finance, National Research Tomsk State University, Russia
M. Quincampoix
Affiliation:
Laboratoire de Mathématiques de Bretagne Atlantique, CNRS–UMR 6205, Université de Bretagne Occidentale, 6 avenue Victor Le Gorgeu, CS 93837, 29238 Brest Cedex 3, France ([email protected])
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Extract

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.

Keywords

boundary-value problemsstochastic averaging methodGreen functions

MSC classification

Secondary: 60H10: Stochastic ordinary differential equations 60J60: Diffusion processes 58J37: Perturbations; asymptotics 34B05: Linear boundary value problems 34C29: Averaging method 34D15: Singular perturbations 34B27: Green functions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 4 , August 2017 , pp. 763 - 779
Copyright
Copyright © Royal Society of Edinburgh 2017 

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