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AIMS Mathematics

2017, Volume 2, Issue 1: 70-80. doi: 10.3934/Math.2017.1.70
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Research article

A Probabilistic Characterization of g-Harmonic Functions

  • 1.
    School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
  • 2.
    Department of Basic Science, Beijing Institute of Graphic Communication, Beijing 102600, China
  • 3.
    Beijing Institute of Education, Beijing 100120, China
  • Received: 04 December 2016 Accepted: 20 December 2016 Published: 05 January 2017

  • Associated with a quasi-linear generator function g, we give a definition of g-harmonic functions. The relation between the g-harmonic functions and g-martingales will be delineated. It is direct to construct such relation for smooth case, but for continuous case we need the theory of viscosity solution. Under the nonlinear expectation mechanism, we can also get the similar relation between harmonic functions and martingales. The strict converse problem of mean value property of g-harmonic functions are discussed finally.

    Citation: Liang Cai, Huan-Huan Zhang, Li-Yun Pan. A Probabilistic Characterization of g-Harmonic Functions[J]. AIMS Mathematics, 2017, 2(1): 70-80. doi: 10.3934/Math.2017.1.70

    Related Papers:

  • Associated with a quasi-linear generator function g, we give a definition of g-harmonic functions. The relation between the g-harmonic functions and g-martingales will be delineated. It is direct to construct such relation for smooth case, but for continuous case we need the theory of viscosity solution. Under the nonlinear expectation mechanism, we can also get the similar relation between harmonic functions and martingales. The strict converse problem of mean value property of g-harmonic functions are discussed finally.


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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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