### Abstract

We prove the existence of two fundamental solutions Φ and Φ̃ of the PDE F(D^{2}Phi) = 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(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}Phi) = 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.

Original language | English (US) |
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Pages (from-to) | 737-777 |

Number of pages | 41 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 64 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2011 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications on Pure and Applied Mathematics*,

*64*(6), 737-777. https://doi.org/10.1002/cpa.20360