The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator

Previous Article
Invariant foliations for stochastic partial differential equations with dynamic boundary conditions
DCDS Home
This Issue
Next Article
Global existence for the stochastic Degasperis-Procesi equation

November 2015, 35(11): 5185-5202. doi: 10.3934/dcds.2015.35.5185

The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator

Laurent Denis ^1, , Anis Matoussi ^1, and Jing Zhang ^2,

1.	Institut du Risque et de l'Assurance, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France, France
2.	School of Mathematical Sciences, University of Fudan, Handan Road 220, 200433, Shanghai, China

Received November 2012 Revised May 2014 Published May 2015

Abstract
Related Papers

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of Itô's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.

Keywords: obstacle problem, space-time white noise., stochastic partial differential equations, Itô's formula, comparison theorem, regular measure, penalization method, non-homogeneous second order operator, Parabolic potential.

Mathematics Subject Classification: 60H15, 35R60, 31B15.

Citation: Laurent Denis, Anis Matoussi, Jing Zhang. The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5185-5202. doi: 10.3934/dcds.2015.35.5185

References:

[1]	D. G. Aronson, On the Green's function for second order parabolic differential equations with discontinuous coefficients, Bulletin of the American Mathematical Society, 69 (1963), 841-847. doi: 10.1090/S0002-9904-1963-11059-9.
[2]	D. G. Aronson, Non-negative solutions of linear parabolic equations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3, 22 (1968), 607-694.
[3]	V. Bally and A. Matoussi, Weak solutions for SPDE's and Backward Doubly SDE's, Journal of Theoret. Probab., 14 (2001), 125-164. doi: 10.1023/A:1007825232513.
[4]	P. Charrier and G. M. Troianiello, Un résultat d'existence et de régularité pour les solutions fortes d'un problème unilatéral d'évolution avec obstacle dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 281 (1975), A621-A623.
[5]	L. Denis and L. Stoïca, A general analytical result for non-linear SPDE's and applications, Electronic Journal of Probability, 9 (2004), 674-709. doi: 10.1214/EJP.v9-223.
[6]	L. Denis, A. Matoussi and L. Stoïca, $L^p$ estimates for the uniform norm of solutions of quasilinear SPDE's, Probability Theory Related Fields, 133 (2005), 437-463. doi: 10.1007/s00440-005-0436-5.
[7]	L. Denis, A. Matoussi and L. Stoïca, Maximum Principle for Parabolic SPDE's: First Approach, Stochastic Partial Differential Equations and Applications VIII in the series Quaderni di Matematica del Dipartimento di Matematica della Seconda Università di Napoli, 2011.
[8]	L. Denis, A. Matoussi and L. Stoïca, Maximum principle and comparison theorem for quasi-linear stochastic PDE's, Electronic Journal of Probability, 14 (2009), 500-530. doi: 10.1214/EJP.v14-629.
[9]	L. Denis and A. Matoussi, Maximum Principle for quasilinear SPDE's on a bounded domain without regularity assumptions, Stochastic Processes and Their Applications, 123 (2013), 1104-1137. doi: 10.1016/j.spa.2012.10.005.
[10]	L. Denis, A. Matoussi and J. Zhang, The obstacle problem for quasilinear stochastic PDEs: Analytical approach, The Annals of Probability, 42 (2014), 865-905. doi: 10.1214/12-AOP805.
[11]	L. Denis, A. Matoussi and J. Zhang, Maximum principle for quasilinear SPDEs with obstacle, Electronic Journal of Probability, 19 (2014), 1-32. doi: 10.1214/EJP.v19-2716.
[12]	C. Donati-Martin and E. Pardoux, White noise driven SPDEs with reflection, Probability Theory Related Fields, 95 (1993), 1-24. doi: 10.1007/BF01197335.
[13]	N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE and related obstacle problems for PDEs, The Annals of Probability, 25 (1997), 702-737. doi: 10.1214/aop/1024404416.
[14]	N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.
[15]	K. H. Kim, An Lp-theory of SPDEs of divergence form on Lipschitz domains, Journal of Theoretical Probability, 22 (2009), 220-238. doi: 10.1007/s10959-008-0170-x.
[16]	T. Klimsiak, Reflected BSDEs and obstacle problem for semilinear PDEs in divergence form, Stochastic Processes and their Applications, 122 (2012), 134-169. doi: 10.1016/j.spa.2011.10.001.
[17]	N. V. Krylov, An analytic approach to SPDEs, in Stochastic Partial Differential Equations: Six Perspectives, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, 1999, 185-242. doi: 10.1090/surv/064/05.
[18]	N. V. Krylov, Maximum principle of SPDEs and its applications, in Stochastic Differential Equations: Theory and Applications (eds. P. Baxendale and S. Lototsky), Interdiscip. Math. Sci., 2, World Scientific, Hackensack, NJ, 2007, 311-338. doi: 10.1142/9789812770639_0012.
[19]	J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Dunod, Paris, 1968.
[20]	A. Matoussi and M. Xu, Sobolev solution for semilinear PDE with obstacle under monotonicity condition, Electronic Journal of Probability, 13 (2008), 1035-1067. doi: 10.1214/EJP.v13-522.
[21]	A. Matoussi and L. Stoïca, The obstacle problem for quasilinear stochastic PDE's, The Annals of Probability, 38 (2010), 1143-1179. doi: 10.1214/09-AOP507.
[22]	F. Mignot and J. P. Puel, Inéquations d'évolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi-variationnelles d'évolution, Arch. for Rat. Mech. and Ana., 64 (1977), 59-91. doi: 10.1007/BF00280179.
[23]	D. Nualart and E. Pardoux, White noise driven quasilinear SPDEs with reflection, Probability Theory and Related Fields, 93 (1992), 77-89. doi: 10.1007/BF01195389.
[24]	M. Pierre, Problèmes d'Evolution avec Contraintes Unilaterales et Potentiels Parabolique, Comm. in Partial Differential Equations, 4 (1979), 1149-1197. doi: 10.1080/03605307908820124.
[25]	M. Pierre, Représentant précis d'un potentiel parabolique, in Séminaire de Théorie du Potentiel, Paris, No. 5, Lecture Notes in Math., 814, Springer, Berlin, 1980, 186-228.
[26]	F. Riesz and B. Nagy, Functional Analysis, Dover, New York, 1990.
[27]	M. Sanz and P. Vuillermot, Equivalence and Hölder Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations, Ann. I. H. Poincaré, 39 (2003), 703-742. doi: 10.1016/S0246-0203(03)00015-3.
[28]	T. G. Xu and T. S. Zhang, White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles, Stochatic Processes and Their Applications, 119 (2009), 3453-3470. doi: 10.1016/j.spa.2009.06.005.

show all references

References:

[1]	D. G. Aronson, On the Green's function for second order parabolic differential equations with discontinuous coefficients, Bulletin of the American Mathematical Society, 69 (1963), 841-847. doi: 10.1090/S0002-9904-1963-11059-9.
[2]	D. G. Aronson, Non-negative solutions of linear parabolic equations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3, 22 (1968), 607-694.
[3]	V. Bally and A. Matoussi, Weak solutions for SPDE's and Backward Doubly SDE's, Journal of Theoret. Probab., 14 (2001), 125-164. doi: 10.1023/A:1007825232513.
[4]	P. Charrier and G. M. Troianiello, Un résultat d'existence et de régularité pour les solutions fortes d'un problème unilatéral d'évolution avec obstacle dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 281 (1975), A621-A623.
[5]	L. Denis and L. Stoïca, A general analytical result for non-linear SPDE's and applications, Electronic Journal of Probability, 9 (2004), 674-709. doi: 10.1214/EJP.v9-223.
[6]	L. Denis, A. Matoussi and L. Stoïca, $L^p$ estimates for the uniform norm of solutions of quasilinear SPDE's, Probability Theory Related Fields, 133 (2005), 437-463. doi: 10.1007/s00440-005-0436-5.
[7]	L. Denis, A. Matoussi and L. Stoïca, Maximum Principle for Parabolic SPDE's: First Approach, Stochastic Partial Differential Equations and Applications VIII in the series Quaderni di Matematica del Dipartimento di Matematica della Seconda Università di Napoli, 2011.
[8]	L. Denis, A. Matoussi and L. Stoïca, Maximum principle and comparison theorem for quasi-linear stochastic PDE's, Electronic Journal of Probability, 14 (2009), 500-530. doi: 10.1214/EJP.v14-629.
[9]	L. Denis and A. Matoussi, Maximum Principle for quasilinear SPDE's on a bounded domain without regularity assumptions, Stochastic Processes and Their Applications, 123 (2013), 1104-1137. doi: 10.1016/j.spa.2012.10.005.
[10]	L. Denis, A. Matoussi and J. Zhang, The obstacle problem for quasilinear stochastic PDEs: Analytical approach, The Annals of Probability, 42 (2014), 865-905. doi: 10.1214/12-AOP805.
[11]	L. Denis, A. Matoussi and J. Zhang, Maximum principle for quasilinear SPDEs with obstacle, Electronic Journal of Probability, 19 (2014), 1-32. doi: 10.1214/EJP.v19-2716.
[12]	C. Donati-Martin and E. Pardoux, White noise driven SPDEs with reflection, Probability Theory Related Fields, 95 (1993), 1-24. doi: 10.1007/BF01197335.
[13]	N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE and related obstacle problems for PDEs, The Annals of Probability, 25 (1997), 702-737. doi: 10.1214/aop/1024404416.
[14]	N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.
[15]	K. H. Kim, An Lp-theory of SPDEs of divergence form on Lipschitz domains, Journal of Theoretical Probability, 22 (2009), 220-238. doi: 10.1007/s10959-008-0170-x.
[16]	T. Klimsiak, Reflected BSDEs and obstacle problem for semilinear PDEs in divergence form, Stochastic Processes and their Applications, 122 (2012), 134-169. doi: 10.1016/j.spa.2011.10.001.
[17]	N. V. Krylov, An analytic approach to SPDEs, in Stochastic Partial Differential Equations: Six Perspectives, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, 1999, 185-242. doi: 10.1090/surv/064/05.
[18]	N. V. Krylov, Maximum principle of SPDEs and its applications, in Stochastic Differential Equations: Theory and Applications (eds. P. Baxendale and S. Lototsky), Interdiscip. Math. Sci., 2, World Scientific, Hackensack, NJ, 2007, 311-338. doi: 10.1142/9789812770639_0012.
[19]	J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Dunod, Paris, 1968.
[20]	A. Matoussi and M. Xu, Sobolev solution for semilinear PDE with obstacle under monotonicity condition, Electronic Journal of Probability, 13 (2008), 1035-1067. doi: 10.1214/EJP.v13-522.
[21]	A. Matoussi and L. Stoïca, The obstacle problem for quasilinear stochastic PDE's, The Annals of Probability, 38 (2010), 1143-1179. doi: 10.1214/09-AOP507.
[22]	F. Mignot and J. P. Puel, Inéquations d'évolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi-variationnelles d'évolution, Arch. for Rat. Mech. and Ana., 64 (1977), 59-91. doi: 10.1007/BF00280179.
[23]	D. Nualart and E. Pardoux, White noise driven quasilinear SPDEs with reflection, Probability Theory and Related Fields, 93 (1992), 77-89. doi: 10.1007/BF01195389.
[24]	M. Pierre, Problèmes d'Evolution avec Contraintes Unilaterales et Potentiels Parabolique, Comm. in Partial Differential Equations, 4 (1979), 1149-1197. doi: 10.1080/03605307908820124.
[25]	M. Pierre, Représentant précis d'un potentiel parabolique, in Séminaire de Théorie du Potentiel, Paris, No. 5, Lecture Notes in Math., 814, Springer, Berlin, 1980, 186-228.
[26]	F. Riesz and B. Nagy, Functional Analysis, Dover, New York, 1990.
[27]	M. Sanz and P. Vuillermot, Equivalence and Hölder Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations, Ann. I. H. Poincaré, 39 (2003), 703-742. doi: 10.1016/S0246-0203(03)00015-3.
[28]	T. G. Xu and T. S. Zhang, White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles, Stochatic Processes and Their Applications, 119 (2009), 3453-3470. doi: 10.1016/j.spa.2009.06.005.

[1]	Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296
[2]	Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure and Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607
[3]	Ying Hu, Shanjian Tang. Nonlinear backward stochastic evolutionary equations driven by a space-time white noise. Mathematical Control and Related Fields, 2018, 8 (3&4) : 739-751. doi: 10.3934/mcrf.2018032
[4]	Shitao Liu, Roberto Triggiani. Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5217-5252. doi: 10.3934/dcds.2013.33.5217
[5]	Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427
[6]	Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[7]	Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077
[8]	Tianlong Shen, Jianhua Huang, Caibin Zeng. Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1523-1533. doi: 10.3934/dcdsb.2018056
[9]	Henri Schurz. Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D. Conference Publications, 2013, 2013 (special) : 673-684. doi: 10.3934/proc.2013.2013.673
[10]	Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353
[11]	Corentin Audiard. On the non-homogeneous boundary value problem for Schrödinger equations. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3861-3884. doi: 10.3934/dcds.2013.33.3861
[12]	Zhen-Zhen Tao, Bing Sun. Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022080
[13]	Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure and Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761
[14]	Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034
[15]	Georgios T. Kossioris, Georgios E. Zouraris. Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1845-1872. doi: 10.3934/dcdsb.2013.18.1845
[16]	Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223
[17]	Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[18]	Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104
[19]	Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
[20]	Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Laurent Denis Anis Matoussi Jing Zhang
on Google Scholar
Laurent Denis Anis Matoussi Jing Zhang

[Back to Top]