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

[1]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

[2]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[3]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[4]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[5]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[6]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[7]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[8]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[9]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[10]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[11]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[12]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[13]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[14]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[15]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[16]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[17]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[18]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[19]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[20]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (56)
  • HTML views (126)
  • Cited by (7)

Other articles
by authors

[Back to Top]