Fundamental solutions of homogeneous fully nonlinear elliptic equations

We prove the existence of two fundamental solutions Φ and $\tilde \Phi$ of the PDE \input amssym $$F(D^2\Phi) = 0 \quad {\rm in} \ {\Bbb{R}}^n \setminus \{ 0 \}$$ for any positively homogeneous, uniformly elliptic operator F . Corresponding to F are two unique scaling exponents α*, $\tilde\alpha^*$ > −1 that describe the homogeneity of Φ 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 2 u ) = 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 2 u ) = 0 in \input amssym ${\Bbb{R}}^n \setminus \{ 0 \}$ that are bounded on one side in both 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. © 2010 Wiley Periodicals, Inc.