Small noise asymptotics for invariant densities for a class of diffusions: A control theoretic view |
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Authors: | Anup Biswas Vivek S. Borkar |
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Affiliation: | aCentre for Applicable Mathematics, Tata Institute of Fundamental Research, Post Bag No. 03, Sharadanagar, Chikkabommasandra, Bangalore 560065, India;bSchool of Technology and Computer Science, Tata Institute of Fundamental Research, Homi Bhabha Rd., Mumbai 400005, India |
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Abstract: | We consider multi-dimensional nondegenerate diffusions with invariant densities, with the diffusion matrix scaled by a small >0. The o.d.e. limit corresponding to =0 is assumed to have the origin as its unique globally asymptotically stable equilibrium. Using control theoretic methods, we show that in the ↓0 limit, the invariant density has the form ≈exp(−W(x)/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. |
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Keywords: | Diffusions Invariant density Small noise limit Hamilton– Jacobi equation Viscosity solution |
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