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http://dx.doi.org/10.4134/BKMS.b180445

SOME STABILITY RESULTS FOR SEMILINEAR STOCHASTIC HEAT EQUATION DRIVEN BY A FRACTIONAL NOISE  

El Barrimi, Oussama (Department of Mathematics Faculty of Sciences Semlalia Cadi Ayyad University)
Ouknine, Youssef (Hassan II Academy of Sciences and Technology Rabat)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.3, 2019 , pp. 631-648 More about this Journal
Abstract
In this paper, we consider a semilinear stochastic heat equation driven by an additive fractional white noise. Under the pathwise uniqueness property, we establish various strong stability results. As a consequence, we give an application to the convergence of the Picard successive approximation.
Keywords
stochastic heat equation; fractional noise; pathwise uniqueness;
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1 G. Denk, D. Meintrup, and S. Schaffer, Modeling, simulation and optimization of integrated circuits, Intern. Ser. Numerical Math. 146 (2004), 251-267.
2 E. Alos and D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stoch. Stoch. Rep. 75 (2003), no. 3, 129-152.   DOI
3 K. Bahlali, M. Eddahbi, and M. Mellouk, Stability and genericity for spde's driven by spatially correlated noise, J. Math. Kyoto Univ. 48 (2008), no. 4, 699-724.   DOI
4 O. El Barrimi and Y. Ouknine, Approximation of solutions of SDEs driven by a fractional Brownian motion, under pathwise uniqueness, Mod. Stoch. Theory Appl. 3 (2016), no. 4, 303-313.   DOI
5 M. Gubinelli, A. Lejay, and S. Tindel, Young integrals and SPDEs, Potential Anal. 25 (2006), no. 4, 307-326.   DOI
6 I. Gyongy and D. Nualart, Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise, Potential Anal. 7 (1997), no. 4, 725-757.   DOI
7 S. C. Kou and X. Sunney, Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule, Phys. Rev. Letters 93 (2004), no. 18.
8 B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422-437.   DOI
9 J. Memin, Y. Mishura, and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 51 (2001), no. 2, 197-206.   DOI
10 D. Nualart and Y. Ouknine, Regularization of quasilinear heat equations by a fractional noise, Stoch. Dyn. 4 (2004), no. 2, 201-221.   DOI
11 L. Quer-Sardanyons and S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet, Stochastic Process. Appl. 117 (2007), no. 10, 1448-1472.   DOI
12 J. M. Rassias, Counterexamples in Differential Equations and Related Topics, World Scientific Publishing Co., Inc., Teaneck, NJ, 1991.
13 V. Bally, I. Gyongy, and Pardoux, White noise driven parabolic SPDEs with measurable drift, J. Funct. Anal. 120 (1994), no. 2, 484-510.   DOI
14 A. V. Skorokhod, Studies in the Theory of Random Processes, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, MA, 1965.
15 J. B. Walsh, An introduction to stochastic partial differential equations, in Ecole d'ete de probabilites de Saint-Flour, XIV-1984, 265-439, Lecture Notes in Math., 1180, Springer, Berlin, 1986.
16 M. Yor, Le drap brownien comme limite en loi de temps locaux lineaires, in Seminar on probability, XVII, 89-105, Lecture Notes in Math., 986, Springer, Berlin, 1983.
17 K. Bahlali, B. Mezerdi, and Y. Ouknine, Pathwise uniqueness and approximation of solutions of stochastic differential equations, in Seminaire de Probabilites, XXXII, 166-187, Lecture Notes in Math., 1686, Springer, Berlin, 1998.
18 R. M. Balan and C. A. Tudor, Erratum to: "The stochastic heat equation with fractionalcolored noise: existence of the solution", ALEA Lat. Am. J. Probab. Math. Stat. 6 (2009), 343-347.
19 V. Bally, A. Millet, and M. Sanz-Sole, Approximation and support theorem in Holder norm for parabolic stochastic partial differential equations, Ann. Probab. 23 (1995), no. 1, 178-222.   DOI
20 X. Bardina, M. Jolis, and L. Quer-Sardanyons, Weak convergence for the stochastic heat equation driven by Gaussian white noise, Electron. J. Probab. 15 (2010), no. 39, 1267-1295.   DOI
21 R. Carmona and D. Nualart, Random nonlinear wave equations: smoothness of the solutions, Probab. Theory Related Fields 79 (1988), no. 4, 469-508.   DOI
22 E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York, 1955.
23 R. C. Dalang and M. Sanz-Sole, Regularity of the sample paths of a class of second-order spde's, J. Funct. Anal. 227 (2005), no. 2, 304-337.   DOI
24 L. Decreusefond and A. S. Ustunel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1999), no. 2, 177-214.   DOI