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

RAMI ATAR; Amarjit Budhiraja

doi:10.1007/s00440-010-0306-7

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

RAMI ATAR, Amarjit Budhiraja

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

A two-player stochastic differential game representation has recently been obtained for solutions of the equation -Δ_∞u = h in a C² domain with Dirichlet boundary condition, where h is continuous and takes values in ℝ\{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.

Original language	English
Pages (from-to)	509-528
Number of pages	20
Journal	Probability Theory and Related Fields
Volume	151
Issue number	3-4
DOIs	https://doi.org/10.1007/s00440-010-0306-7
State	Published - Dec 2011

Keywords

Bellman-Issacs equations
Infinity-Laplacian
Stochastic differential games

All Science Journal Classification (ASJC) codes

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

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10.1007/s00440-010-0306-7

Cite this

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title = "On near optimal trajectories for a game associated with the ∞-Laplacian",

abstract = "A two-player stochastic differential game representation has recently been obtained for solutions of the equation -Δ∞u = h in a C2 domain with Dirichlet boundary condition, where h is continuous and takes values in ℝ\{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.",

keywords = "Bellman-Issacs equations, Infinity-Laplacian, Stochastic differential games",

author = "RAMI ATAR and Amarjit Budhiraja",

note = "Funding 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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doi = "10.1007/s00440-010-0306-7",

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AU - ATAR, RAMI

AU - Budhiraja, Amarjit

N1 - Funding 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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N2 - A two-player stochastic differential game representation has recently been obtained for solutions of the equation -Δ∞u = h in a C2 domain with Dirichlet boundary condition, where h is continuous and takes values in ℝ\{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.

AB - A two-player stochastic differential game representation has recently been obtained for solutions of the equation -Δ∞u = h in a C2 domain with Dirichlet boundary condition, where h is continuous and takes values in ℝ\{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.

KW - Bellman-Issacs equations

KW - Infinity-Laplacian

KW - Stochastic differential games

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U2 - 10.1007/s00440-010-0306-7

DO - 10.1007/s00440-010-0306-7

M3 - مقالة

SN - 0178-8051

VL - 151

SP - 509

EP - 528

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -

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

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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