Small noise asymptotics for invariant densities for a class of diffusions: A control theoretic view

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:360

Article

Biswas, Anup²  Borkar, Vivek S.¹ 

[1] Tata Inst Fundamental Res, Sch Technol & Comp Sci, Bombay 400005, Maharashtra, India

[2] Tata Inst Fundamental Res, Ctr Appl Math, Bangalore 560065, Karnataka, India

关键词: Diffusions;    Invariant density;    Small noise limit;    Hamilton-Jacobi equation;    Viscosity solution;   

DOI  :  10.1016/j.jmaa.2009.06.070

来源: Elsevier

【 摘 要 】

We consider multi-dimensional nondegenerate diffusions with invariant densities, with the diffusion matrix scaled by a small epsilon > 0. The o.d.e. limit corresponding to epsilon = 0 is assumed to have the origin as its unique globally asymptotically stable equilibrium. Using control theoretic methods, we show that in the epsilon down arrow 0 limit, the invariant density has the form approximate to exp(-W(x)/epsilon(2)), where the W is characterized as the optimal cost of a deterministic control problem. This generalizes an earlier work of Sheu. Extension to multiple equilibria is also given. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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