May  2017, 16(3): 915-944. doi: 10.3934/cpaa.2017044

Tug-of-war games with varying probabilities and the normalized p(x)-laplacian

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

2. 

Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, FI-40014 Jyväskylä, Finland

Received  August 2016 Revised  December 2016 Published  February 2017

We study a two player zero-sum tug-of-war game with varying probabilities that depend on the game location x. In particular, we show that the value of the game is locally asymptotically Hölder continuous. The main difficulty is the loss of translation invariance. We also show the existence and uniqueness of values of the game. As an application, we prove that the value function of the game converges to a viscosity solution of the normalized p(x) -Laplacian.

Citation: Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure & Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044
References:
[1]

T. Adamowicz and P. Hästö, Harnack's inequality and the strong p(·)-Laplacian, J. Differential Equations, 260 (2011), 1631-1649.  doi: 10.1016/j.jde.2010.10.006.  Google Scholar

[2]

S. N. Armstrong and C. K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc., 364 (2012), 595-636.  doi: 10.1090/S0002-9947-2011-05289-X.  Google Scholar

[3]

L. A. Caffarelli and X. Cabré, Fully nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043.  Google Scholar

[4]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature, I, J. Differential Geom., 33 (1991), 635-681.   Google Scholar

[5]

Y. Giga, Surface Evolution Equations. A Level Set Approach, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[6]

H. Hartikainen, A dynamic programming principle with continuous solutions related to the p-Laplacian, 1 < p < ∞, Differential Integral Equations, 29 (2016), 583-600.   Google Scholar

[7]

Ishii and P.-Lions L. H., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[8]

B. KawohlJ. J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188.  doi: 10.1016/j.matpur.2011.07.001.  Google Scholar

[9]

S. Koike, A Beginner's Guide to The Theory of Viscosity Solutions, Mathematical Society of Japan, Tokyo, 2004.  Google Scholar

[10]

S. Kusuoka, Hölder continuity and bounds for fundamental solutions to nondivergence form parabolic equations, Anal. PDE, 8 (2015), 1-32.  doi: 10.2140/apde.2015.8.1.  Google Scholar

[11]

H. Luiro and M. Parviainen, Regularity for nonlinear stochastic games, preprint, arXiv: 1509.07263. Google Scholar

[12]

H. LuiroM. Parviainen and E. Saksman, Harnack's inequality for p-harmonic functions via stochastic games, Comm. Partial Differential Equations, 38 (2013), 1985-2003.  doi: 10.1080/03605302.2013.814068.  Google Scholar

[13]

H. LuiroM. Parviainen and E. Saksman, On the existence and uniqueness of p-harmonious functions, Differential Integral Equations, 27 (2014), 201-216.   Google Scholar

[14]

T. Lindvall and L. C. G. Rogers, Coupling of multidimensional diffusions by reflection, Ann. Probab., 14 (1986), 860-872.   Google Scholar

[15]

J. J. ManfrediM. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889.  doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar

[16]

J. J. ManfrediM. Parviainen and J. D. Rossi, On the definition and properties of pharmonious functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (2012), 215-241.   Google Scholar

[17]

Pérez-Llanos M., A homogenization process for the strong p(x)-Laplacian, Nonlinear Anal., 76 (2013), 105-114.  doi: 10.1016/j.na.2012.08.006.  Google Scholar

[18]

A. Porretta and E. Priola, Global Lipschitz regularizing effects for linear and nonlinear parabolic equations, J. Math. Pures Appl., 100 (2013), 633-686.  doi: 10.1016/j.matpur.2013.01.016.  Google Scholar

[19]

Y. Peres and S. Sheffield, Tug-of-war with noise: a game-theoretic view of the p-Laplacian, Duke Math. J., 145 (2008), 91-120.  doi: 10.1215/00127094-2008-048.  Google Scholar

[20]

Y. PeresO. SchrammS. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.  Google Scholar

[21]

E. Priola and F.-Y. Wang, Gradient estimates for diffusion semigroups with singular coefficients, J. Funct. Anal., 236 (2006), 244-264.  doi: 10.1016/j.jfa.2005.12.010.  Google Scholar

[22] S. M. Srivastava, A course on Borel sets, Springer-Verlag, New York, 1998.  doi: 10.1007/978-3-642-85473-6.  Google Scholar
[23]

C. Zhang and S. Zhou, Hölder regularity for the gradients of solutions of the strong p(x)-Laplacian, J. Math. Anal. Appl., 389 (2013), 1066-1077.  doi: 10.1016/j.jmaa.2011.12.047.  Google Scholar

show all references

References:
[1]

T. Adamowicz and P. Hästö, Harnack's inequality and the strong p(·)-Laplacian, J. Differential Equations, 260 (2011), 1631-1649.  doi: 10.1016/j.jde.2010.10.006.  Google Scholar

[2]

S. N. Armstrong and C. K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc., 364 (2012), 595-636.  doi: 10.1090/S0002-9947-2011-05289-X.  Google Scholar

[3]

L. A. Caffarelli and X. Cabré, Fully nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043.  Google Scholar

[4]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature, I, J. Differential Geom., 33 (1991), 635-681.   Google Scholar

[5]

Y. Giga, Surface Evolution Equations. A Level Set Approach, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[6]

H. Hartikainen, A dynamic programming principle with continuous solutions related to the p-Laplacian, 1 < p < ∞, Differential Integral Equations, 29 (2016), 583-600.   Google Scholar

[7]

Ishii and P.-Lions L. H., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[8]

B. KawohlJ. J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188.  doi: 10.1016/j.matpur.2011.07.001.  Google Scholar

[9]

S. Koike, A Beginner's Guide to The Theory of Viscosity Solutions, Mathematical Society of Japan, Tokyo, 2004.  Google Scholar

[10]

S. Kusuoka, Hölder continuity and bounds for fundamental solutions to nondivergence form parabolic equations, Anal. PDE, 8 (2015), 1-32.  doi: 10.2140/apde.2015.8.1.  Google Scholar

[11]

H. Luiro and M. Parviainen, Regularity for nonlinear stochastic games, preprint, arXiv: 1509.07263. Google Scholar

[12]

H. LuiroM. Parviainen and E. Saksman, Harnack's inequality for p-harmonic functions via stochastic games, Comm. Partial Differential Equations, 38 (2013), 1985-2003.  doi: 10.1080/03605302.2013.814068.  Google Scholar

[13]

H. LuiroM. Parviainen and E. Saksman, On the existence and uniqueness of p-harmonious functions, Differential Integral Equations, 27 (2014), 201-216.   Google Scholar

[14]

T. Lindvall and L. C. G. Rogers, Coupling of multidimensional diffusions by reflection, Ann. Probab., 14 (1986), 860-872.   Google Scholar

[15]

J. J. ManfrediM. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889.  doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar

[16]

J. J. ManfrediM. Parviainen and J. D. Rossi, On the definition and properties of pharmonious functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (2012), 215-241.   Google Scholar

[17]

Pérez-Llanos M., A homogenization process for the strong p(x)-Laplacian, Nonlinear Anal., 76 (2013), 105-114.  doi: 10.1016/j.na.2012.08.006.  Google Scholar

[18]

A. Porretta and E. Priola, Global Lipschitz regularizing effects for linear and nonlinear parabolic equations, J. Math. Pures Appl., 100 (2013), 633-686.  doi: 10.1016/j.matpur.2013.01.016.  Google Scholar

[19]

Y. Peres and S. Sheffield, Tug-of-war with noise: a game-theoretic view of the p-Laplacian, Duke Math. J., 145 (2008), 91-120.  doi: 10.1215/00127094-2008-048.  Google Scholar

[20]

Y. PeresO. SchrammS. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.  Google Scholar

[21]

E. Priola and F.-Y. Wang, Gradient estimates for diffusion semigroups with singular coefficients, J. Funct. Anal., 236 (2006), 244-264.  doi: 10.1016/j.jfa.2005.12.010.  Google Scholar

[22] S. M. Srivastava, A course on Borel sets, Springer-Verlag, New York, 1998.  doi: 10.1007/978-3-642-85473-6.  Google Scholar
[23]

C. Zhang and S. Zhou, Hölder regularity for the gradients of solutions of the strong p(x)-Laplacian, J. Math. Anal. Appl., 389 (2013), 1066-1077.  doi: 10.1016/j.jmaa.2011.12.047.  Google Scholar

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