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Global existence for the stochastic Degasperis-Procesi equation
The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator
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 |
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. |
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