  We prove the existence of two fundamental solutions $\Phi$ and $\tilde \Phi$
of the PDE \[ F(D^2\Phi) = 0 \quad {in} \mathbb{R}^n \setminus \{0 \} \] for
any positively homogeneous, uniformly elliptic operator $F$. Corresponding to
$F$ are two unique scaling exponents $\alpha^*, \tilde\alpha^* > -1$ which
describe the homogeneity of $\Phi$ and $\tilde \Phi$. We give a sharp
characterization of the isolated singularities and the behavior at infinity of
a solution of the equation $F(D^2u) = 0$, which is bounded on one side. A
Liouville-type result demonstrates that the two fundamental solutions are the
unique nontrivial solutions of $F(D^2u) = 0$ in $\mathbb{R}^n \setminus \{0 \}$
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.
