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Gradient walks and $ p$-harmonic functions


Authors: Hannes Luiro and Mikko Parviainen
Journal: Proc. Amer. Math. Soc. 145 (2017), 4313-4324
MSC (2010): Primary 35J92, 60H30, 60J05, 60J60
DOI: https://doi.org/10.1090/proc/13540
Published electronically: June 9, 2017
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Abstract: We consider a class of stochastic processes having a connection to $ p$-harmonic functions. In particular, we obtain stochastic approximations that converge uniformly to a $ p$-harmonic function, and also with an explicit convergence rate. The main difficulty is how to deal with the zero set of the gradient of the underlying function.


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Hannes Luiro
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014 Jyväskylä, Finland
Email: hannes.s.luiro@jyu.fi

Mikko Parviainen
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014 Jyväskylä, Finland
Email: mikko.j.parviainen@jyu.fi

DOI: https://doi.org/10.1090/proc/13540
Keywords: Markov chain, stochastic representations for $p$-harmonic functions, tug-of-war with noise
Received by editor(s): May 18, 2016
Received by editor(s) in revised form: September 18, 2016, and October 11, 2016
Published electronically: June 9, 2017
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2017 American Mathematical Society