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2017 | OriginalPaper | Buchkapitel

5. Obstacle Problems

verfasst von : Lorenzo Zambotti

Erschienen in: Random Obstacle Problems

Verlag: Springer International Publishing

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Abstract

In this chapter we introduce the SPDEs with reflection which are studied in detail in the following chapters. The theory at the beginning is purely deterministic: as in the Skorohod Lemma 2.1, we have a driving continuous function (t, x) ↦ w(t, x) which plays the role of an obstacle and we look for a continuous function z ≥ −w which solves a heat equation on the open set {z > −w} and is reflected on − w. Following Nualart and Pardoux [NP92] we give an existence and uniqueness result for solutions to such obstacle problems.

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Vorheriges Kapitel The Stochastic Heat Equation

Nächstes Kapitel Integration by Parts Formulae

ASZ09.

L. Ambrosio, G. Savaré, L. Zambotti, Existence and stability for Fokker-Planck equations with log-concave reference measure. Probab. Theory Relat. Fields 145 (3–4), 517–564 (2009). doi:10.1007/s00440-008-0177-3

BDPT12.

V. Barbu, G. Da Prato, L. Tubaro, The stochastic reflection problem in Hilbert spaces. Commun. Partial Differ. Equ. 37 (2), 352–367 (2012). doi:10.1080/03605302.2011.596878

BL78.

A. Bensoussan, J.-L. Lions, Applications des inéquations variationnelles en contrôle stochastique (Dunod, Paris, 1978), pp. viii+545; Méthodes Mathématiques de l’Informatique, No. 6

Cép98.

E. Cépa, Problème de Skorohod multivoque. Ann. Probab. 26 (2), 500–532 (1998). doi:10.1214/aop/1022855642

Cer01.

S. Cerrai, Second Order PDE’s in Finite and Infinite Dimension. Lecture Notes in Mathematics, vol. 1762 (Springer, Berlin, 2001), pp. x+330. doi:10.1007/b80743

DG11.

A. Debussche, L. Goudenège, Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections. SIAM J. Math. Anal. 43 (3), 1473–1494 (2011). doi:10.1137/090769636

DMP93.

C. Donati-Martin, É. Pardoux, White noise driven SPDEs with reflection. Probab. Theory Relat. Fields 95 (1), 1–24 (1993). doi:10.1007/BF01197335

DMP97.

C. Donati-Martin, E. Pardoux, EDPS réfléchies et calcul de Malliavin. Bull. Sci. Math. 121 (5), 405–422 (1997)MathSciNetMATH

DPZ96.

G. Da Prato, J. Zabczyk, Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series, vol. 229 (Cambridge University Press, Cambridge, 1996), pp. xii+339. doi:10.1017/CBO9780511662829

DZ07.

A. Debussche, L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection. Ann. Probab. 35 (5), 1706–1739 (2007). doi:10.1214/009117906000000773

Eva08.

S.N. Evans, Probability and Real Trees. Lecture Notes in Mathematics, vol. 1920 (Springer, Berlin, 2008), pp. xii+193; Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6–23, 2005. doi:10.1007/978-3-540-74798-7

FO01.

T. Funaki, S. Olla, Fluctuations for ∇ϕ interface model on a wall. Stoch. Process. Appl. 94 (1), 1–27 (2001). doi:10.1016/S0304-4149(00)00104-6

GM15.

L. Goudenège, L. Manca, Asymptotic properties of stochastic Cahn-Hilliard equation with singular nonlinearity and degenerate noise. Stoch. Process. Appl. 125 (10), 3785–3800 (2015) doi:10.1016/j.spa.2015.05.006

Gou09.

L. Goudenège, Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection. Stoch. Process. Appl. 119 (10), 3516–3548 (2009). doi:10.1016/j.spa.2009.06.008

LS84.

P.-L. Lions, A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37 (4), 511–537 (1984). doi:10.1002/cpa.3160370408

NP92.

D. Nualart, É. Pardoux, White noise driven quasilinear SPDEs with reflection. Probab. Theory Relat. Fields 93 (1), 77–89 (1992). doi:10.1007/BF01195389

RZZ12.

M. Röckner, R.-C. Zhu, X.-C. Zhu, The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple. Ann. Probab. 40 (4), 1759–1794 (2012). doi:10.1214/11-AOP661

XZ09.

T. Xu, T. Zhang, White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles. Stoch. Process. Appl. 119 (10), 3453–3470 (2009). doi:10.1016/j.spa.2009.06.005

Zam01.

L. Zambotti, A reflected stochastic heat equation as symmetric dynamics with respect to the 3-d Bessel bridge. J. Funct. Anal. 180 (1), 195–209 (2001). doi:10.1006/jfan.2000.3685

Zam08b.

L. Zambotti, Fluctuations for a conservative interface model on a wall. ALEA Lat. Am. J. Probab. Math. Stat. 4, 167–184 (2008)MathSciNetMATH

Titel: Obstacle Problems
verfasst von: Lorenzo Zambotti
Verlag: Springer International Publishing
Buch: Random Obstacle Problems
Print ISBN: 978-3-319-52095-7

Electronic ISBN: 978-3-319-52096-4

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-52096-4_5