We prove the existence of two fundamental solutions Φ and Φ~
of the PDE F(D2Φ)=0inRn∖{0} for
any positively homogeneous, uniformly elliptic operator F. Corresponding to
F are two unique scaling exponents α∗,α~∗>−1 which
describe the homogeneity of Φ and Φ~. We give a sharp
characterization of the isolated singularities and the behavior at infinity of
a solution of the equation F(D2u)=0, which is bounded on one side. A
Liouville-type result demonstrates that the two fundamental solutions are the
unique nontrivial solutions of F(D2u)=0 in Rn∖{0}
which are bounded on one side in a neighborhood of the origin as well as at
infinity. Finally, we show that the sign of each scaling exponent is related to
the recurrence or transience of a stochastic process for a two-player
differential game.Comment: 35 pages, typos and minor mistakes correcte