Fundamental solutions of homogeneous fully nonlinear elliptic equations

We prove the existence of two fundamental solutions Φ and Φ̃ of the PDE F(D²Phi) = 0 R{double-struck}ⁿ\{0} for any positively homogeneous, uniformly elliptic operator F. Corresponding to F are two unique scaling exponents α*, α̃* > -1 that describe the homogeneity of Φ and Φ̃. 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(D²Phi) = 0 R{double-struck}ⁿ\{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.