The tug-of-war without noise and the infinity Laplacian in a wedge. (English) Zbl 1300.60057
Summary: Consider the ending time of the tug-of-war without noise in a wedge. There is a critical angle for finiteness of its expectation when player I maximizes the distance to the boundary and player II minimizes the distance. There is also a critical angle such that for smaller angles, player II can find a strategy where the expected ending time is finite, regardless of player I’s strategy. For larger angles, for each strategy of player II, player I can find a strategy making the expected ending time infinite. Using connections with the inhomogeneous infinity Laplacian, we bound this critical angle.
MSC:
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 91A15 | Stochastic games, stochastic differential games |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 91A24 | Positional games (pursuit and evasion, etc.) |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 60G42 | Martingales with discrete parameter |
Keywords:
tug-of-war without noise; wedge; inhomogeneous game \(\infty\)-Laplacian; viscosity solution; expected ending time of a game; critical angleReferences:
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