Fundamental solutions of homogeneous fully nonlinear elliptic equations

We prove the existence of two fundamental solutions $\mathrm{\Phi }$ and $\tilde{\mathrm{\Phi }}$ of the PDE $$F(D^2\mathrm{\Phi })=0in{\mathbb{R}}^n\setminus \{0\}$$ for any positively homogeneous, uniformly elliptic operator $F$. Corresponding to $F$ are two unique scaling exponents ${\alpha }^{\ast },{\tilde{\alpha }}^{\ast }>-1$ which describe the homogeneity of $\mathrm{\Phi }$ and $\tilde{\mathrm{\Phi }}$. 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 ${\mathbb{R}}^n\setminus \{0\}$ 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.