On the definition and properties of p-harmonious functions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, p. 215-241

We consider functions that satisfy the identity

uϵ(x)=α2supB¯ϵ(x)uϵ+infB¯ϵ(x)uϵ+β-Bϵ(x)uϵdy

for a bounded domain in n . Here ϵ>0 and α, and β are suitable nonnegative coefficients such that α+β=1. In particular, we show that these functions are uniquely determined by their boundary values, approximate p-harmonic functions for 2p< (for a choice of p that depends on α and β), and satisfy the strong comparison principle. We also analyze their relation to the theory of tug-of-war games with noise.

Published online : 2018-06-21
Classification:  91A15,  35B50,  35J25,  35J70,  49N70,  91A24
@article{ASNSP_2012_5_11_2_215_0,
     author = {Manfredi, Juan J. and Parviainen, Mikko and Rossi, Julio D.},
     title = {On the definition and properties of $p$-harmonious functions},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {2},
     year = {2012},
     pages = {215-241},
     zbl = {1252.91014},
     mrnumber = {3011990},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_2_215_0}
}
Manfredi, Juan J.; Parviainen, Mikko; Rossi, Julio D. On the definition and properties of $p$-harmonious functions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, pp. 215-241. http://www.numdam.org/item/ASNSP_2012_5_11_2_215_0/

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