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 and 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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

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

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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

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

[1]

Ivana Gómez, Julio D. Rossi. Tug-of-war games and the infinity Laplacian with spatial dependence. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1959-1983. doi: 10.3934/cpaa.2013.12.1959

[2]

Juan J. Manfredi, Julio D. Rossi, Stephanie J. Somersille. An obstacle problem for Tug-of-War games. Communications on Pure and Applied Analysis, 2015, 14 (1) : 217-228. doi: 10.3934/cpaa.2015.14.217

[3]

Guangying Lv, Hongjun Gao, Jinlong Wei, Jiang-Lun Wu. On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022119

[4]

Arnulf Jentzen, Benno Kuckuck, Thomas Müller-Gronbach, Larisa Yaroslavtseva. Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3707-3724. doi: 10.3934/dcdsb.2021203

[5]

Samia Challal, Abdeslem Lyaghfouri. Hölder continuity of solutions to the $A$-Laplace equation involving measures. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1577-1583. doi: 10.3934/cpaa.2009.8.1577

[6]

Lili Li, Chunrong Chen. Nonlinear scalarization with applications to Hölder continuity of approximate solutions. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 295-307. doi: 10.3934/naco.2014.4.295

[7]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897

[8]

Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741

[9]

Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375

[10]

Kerstin Does. An evolution equation involving the normalized $P$-Laplacian. Communications on Pure and Applied Analysis, 2011, 10 (1) : 361-396. doi: 10.3934/cpaa.2011.10.361

[11]

Xiaohui Zhang, Xuping Zhang. Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022081

[12]

Jeongmin Han. Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2617-2640. doi: 10.3934/cpaa.2020114

[13]

Jian Song, Meng Wang. Stochastic maximum principle for systems driven by local martingales with spatial parameters. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 213-236. doi: 10.3934/puqr.2021011

[14]

Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22

[15]

Said Taarabti. Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 229-243. doi: 10.3934/dcdss.2021029

[16]

Mohamed Karim Hamdani, Lamine Mbarki, Mostafa Allaoui, Omar Darhouche, Dušan D. Repovš. Existence and multiplicity of solutions involving the $ p(x) $-Laplacian equations: On the effect of two nonlocal terms. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022129

[17]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553

[18]

Michela Eleuteri, Paolo Marcellini, Elvira Mascolo. Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 251-265. doi: 10.3934/dcdss.2019018

[19]

Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial and Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27

[20]

Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics and Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (183)
  • HTML views (134)
  • Cited by (8)

Other articles
by authors

[Back to Top]