### Abstract

In this paper we investigate a class of harmonic functions associated with a pair x_{t} = (x_{t1}^{1}, x_{t2}^{2}) of strong Markov processes. In the case where both processes are Brownian motions, a smooth function f is harmonic if Δ_{x1}Δ_{x2}f(x^{1 },x^{2}) = 0. For these harmonic functions we investigate a certain boundary value problem which is analogous to the Dirichlet problem associated with a single process. One basic tool for this study is a generalization of Dynkin's formula, which can be thought of as a kind of stochastic Green's formula. Another important tool is the use of Markov processes x_{ti}^{-i} obtained from x_{ti}^{i} by certain random time changes. We call such a process a stochastic wave since it propogates deterministically through a certain family of sets; however its position on a given set is random.

Original language | English (US) |
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Pages (from-to) | 125-144 |

Number of pages | 20 |

Journal | Advances in Applied Mathematics |

Volume | 4 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1983 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Applied Mathematics