AIMS Mathematics, 2017, 2(1): 70-80. doi: 10.3934/Math.2017.1.70

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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

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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