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On near optimal trajectories for a game associated with the ∞-Laplacian
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  • Published: 09 June 2010

On near optimal trajectories for a game associated with the ∞-Laplacian

  • Rami Atar1 &
  • Amarjit Budhiraja2 

Probability Theory and Related Fields volume 151, pages 509–528 (2011)Cite this article

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  • 2 Citations

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Abstract

A two-player stochastic differential game representation has recently been obtained for solutions of the equation −Δ∞ u = h in a \({{\mathcal C}^2}\) domain with Dirichlet boundary condition, where h is continuous and takes values in \({{\mathbb R}{\setminus}\{0\}}\) . Under appropriate assumptions, including smoothness of u, we identify a family of diffusion processes that may arise as the vanishing δ limit law of the state process, when both players play δ-optimally. We also identify the limit law of the state process under a sequence of near saddle points.

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Rami Atar

  2. Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC, 27599, USA

    Amarjit Budhiraja

Authors
  1. Rami Atar
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  2. Amarjit Budhiraja
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Corresponding author

Correspondence to Amarjit Budhiraja.

Additional information

Research of R. Atar and A. Budhiraja was supported in part by the US-Israel Binational Science Foundation (Grant 2008466), research of R. Atar was supported in part by the Israel Science Foundation (Grant 1349/08) and research of A. Budhiraja was supported in part by the Army Research Office (Grant W911NF-0-1-0080).

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Atar, R., Budhiraja, A. On near optimal trajectories for a game associated with the ∞-Laplacian. Probab. Theory Relat. Fields 151, 509–528 (2011). https://doi.org/10.1007/s00440-010-0306-7

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  • Received: 01 December 2008

  • Revised: 12 May 2010

  • Published: 09 June 2010

  • Issue Date: December 2011

  • DOI: https://doi.org/10.1007/s00440-010-0306-7

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Keywords

  • Stochastic differential games
  • Infinity-Laplacian
  • Bellman-Issacs equations

Mathematics Subject Classification (2000)

  • 91A15
  • 91A23
  • 35J70
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