 
Summary: The Annals of Probability
2010, Vol. 38, No. 2, 498531
DOI: 10.1214/09AOP494
© Institute of Mathematical Statistics, 2010
A STOCHASTIC DIFFERENTIAL GAME FOR THE
INHOMOGENEOUS LAPLACE EQUATION
BY RAMI ATAR AND AMARJIT BUDHIRAJA1
TechnionIsrael Institute of Technology and University of North Carolina
Given a bounded C2 domain G Rm, functions g C(G,R) and
h C(G,R \ {0}), let u denote the unique viscosity solution to the equation
2 u = h in G with boundary data g. We provide a representation for u
as the value of a twoplayer zerosum stochastic differential game.
1. Introduction.
1.1. InfinityLaplacian and games. For an integer m 2, let a bounded C2 do
main G Rm, functions g C(G,R) and h C(G,R \ {0}) be given. We study
a twoplayer zerosum stochastic differential game (SDG), defined in terms of an
mdimensional state process that is driven by a onedimensional Brownian motion,
played until the state exits the domain. The functions g and h serve as terminal,
and, respectively, running payoffs. The players' controls enter in a diffusion coef
ficient and in an unbounded drift coefficient of the state process. The dynamics are
