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Exit time and invariant measure asymptotics for small noise constrained diffusions

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  • Biswas, Anup
  • Budhiraja, Amarjit

Abstract

Constrained diffusions, with diffusion matrix scaled by small ϵ>0, in a convex polyhedral cone G⊂Rk, are considered. Under suitable stability assumptions small noise asymptotic properties of invariant measures and exit times from domains are studied. Let B⊂G be a bounded domain. Under conditions, an “exponential leveling” property that says that, as ϵ→0, the moments of functionals of exit location from B, corresponding to distinct initial conditions, coalesce asymptotically at an exponential rate, is established. It is shown that, with appropriate conditions, difference of moments of a typical exit time functional with a sub-logarithmic growth, for distinct initial conditions in suitable compact subsets of B, is asymptotically bounded. Furthermore, as initial conditions approach 0 at a rate ϵ2 these moments are shown to asymptotically coalesce at an exponential rate.

Suggested Citation

  • Biswas, Anup & Budhiraja, Amarjit, 2011. "Exit time and invariant measure asymptotics for small noise constrained diffusions," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 899-924.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:5:p:899-924
    DOI: 10.1016/j.spa.2011.01.006
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    References listed on IDEAS

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    1. Budhiraja, Amarjit & Lee, Chihoon, 2007. "Long time asymptotics for constrained diffusions in polyhedral domains," Stochastic Processes and their Applications, Elsevier, vol. 117(8), pages 1014-1036, August.
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    3. Borkar, V. S., 2003. "Dynamic programming for ergodic control with partial observations," Stochastic Processes and their Applications, Elsevier, vol. 103(2), pages 293-310, February.
    4. Dupuis, Paul & Ramanan, Kavita, 2002. "A time-reversed representation for the tail probabilities of stationary reflected Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 98(2), pages 253-287, April.
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    Cited by:

    1. Amarjit Budhiraja & Jiang Chen & Sylvain Rubenthaler, 2014. "A Numerical Scheme for Invariant Distributions of Constrained Diffusions," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 262-289, May.
    2. Salins, M., 2021. "Systems of small-noise stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over all initial data," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 159-194.

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