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