Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions

  • Siva R. Athreya
  • Vivek S. Borkar
  • K. Suresh Kumar
  • Rajesh SundaresanEmail author


We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by
$$\begin{aligned} dX^{\varepsilon }_t= & {} b(X^{\varepsilon }_t, Y^{\varepsilon }_t)dt + \varepsilon ^{\alpha }dB_t, \\ dY^{\varepsilon }_t= & {} - \frac{1}{\varepsilon } \nabla _yU(X^{\varepsilon }_t, Y^{\varepsilon }_t)dt + \frac{s(\varepsilon )}{\sqrt{\varepsilon }} dW_t, \end{aligned}$$
where \(B_t, W_t\) are independent Brownian motions on \({\mathbb R}^d\) and \({\mathbb R}^m\) respectively, \(b : \mathbb {R}^d \times \mathbb {R}^m \rightarrow \mathbb {R}^d\), \(U : \mathbb {R}^d \times \mathbb {R}^m \rightarrow \mathbb {R}\) and \(s :(0,\infty ) \rightarrow (0,\infty )\). We impose regularity assumptions on b, U and let \(0< \alpha < 1.\) When \(s(\varepsilon )\) goes to zero slower than a prescribed rate as \(\varepsilon \rightarrow 0\), we characterize all weak limit points of \(X^{\varepsilon }\), as \(\varepsilon \rightarrow 0\), as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of \(U(x, \cdot )\) at its global minima we characterize all limit points as Filippov solutions to the differential equation.


Averaging principle Slow–fast motion Carathéodory solution Filippov solution Small noise limit Nonlinear filter Spectral gap Reversible diffusion 

Mathematics Subject Classification

60J60 60G35 



Research of S.R.A. was supported in part by ISF-UGC grant, research of V.S.B. was supported in part by a J. C. Bose Fellowship, research of K.S.K. was supported in part by the grant MTR/2017/000416 from SERB and research of R.S. was supported in part by RBCCPS-IISc. S.R.A., V.S.B. and R.S. would like to thank the International Centre for Theoretical Sciences (ICTS) for hospitality during the Large deviation theory in statistical physics: Recent advances and future challenges (Code:ICTS/Prog-ldt/2017/8). The authors thank Sanjoy Mitter for pointing out the reference [29], Laurent Miclo and Patrick Cattiaux for suggestions on the spectral gap estimate in Proposition 2.1(b), and Konstantinos Spiliopoulos for pointing out several references in the literature.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Stat-Math UnitIndian Statistical InstituteBangaloreIndia
  2. 2.Department of Electrical EngineeringIndian Institute of TechnologyMumbaiIndia
  3. 3.Department of MathematicsIndian Institute of TechnologyMumbaiIndia
  4. 4.Department of Electrical Communication Engineering and Robert Bosch Centre for Cyber-Physical SystemsIndian Institute of ScienceBangaloreIndia

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