Applications of Mathematics, Vol. 63, No. 1, pp. 7-35, 2018
Stochastic affine evolution equations with multiplicative fractional noise
Bohdan Maslowski, Jana Šnupárková
Received February 13, 2017. First published January 8, 2018.
Abstract: A stochastic affine evolution equation with bilinear noise term is studied, where the driving process is a real-valued fractional Brownian motion with Hurst parameter greater than $1/2$. Stochastic integration is understood in the Skorokhod sense. The existence and uniqueness of weak solution is proved and some results on the large time dynamics are obtained.
References: [1] E. Alòs, D. Nualart: Stochastic integration with respect to the fractional Brownian motion. Stochastics Stochastics Rep. 75 (2002), 129-152. DOI 10.1080/1045112031000078917 | MR 1978896 | Zbl 1028.60048
[2] J. Bártek, M. J. Garrido-Atienza, B. Maslowski: Stochastic porous media equation driven by fractional Brownian motion. Stoch. Dyn. 13 (2013), Article ID 1350010, 33 pages. DOI 10.1142/S021949371350010X | MR 3116928 | Zbl 1291.35453
[3] S. Bonaccorsi: Nonlinear stochastic differential equations in infinite dimensions. Stochastic Anal. Appl. 18 (2000), 333-345. DOI 10.1080/07362990008809673 | MR 1758177 | Zbl 0959.60046
[4] G. Da Prato, M. Iannelli, L. Tubaro: An existence result for a linear abstract stochastic equation in Hilbert spaces. Rend. Sem. Mat. Univ. Padova 67 (1982), 171-180. MR 0682709 | Zbl 0499.60061
[5] G. Da Prato, M. Iannelli, L. Tubaro: Some results on linear stochastic differential equations in Hilbert spaces. Stochastics 6 (1982), 105-116. DOI 10.1080/17442508208833210 | MR 0665246 | Zbl 0475.60041
[6] G. Da Prato, J. Zabczyk: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44, Cambridge University Press, Cambridge (1992). DOI 10.1017/CBO9780511666223 | MR 1207136 | Zbl 0761.60052
[7] T. E. Duncan, B. Maslowski, B. Pasik-Duncan: Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stochastic Processes Appl. 115 (2005), 1357-1383. DOI 10.1016/j.spa.2005.03.011 | MR 2152379 | Zbl 1076.60054
[8] X. Fernique: Régularité des trajectoires des fonctions aléatoires gaussiennes. Ec. d'Ete Probab. Saint-Flour IV-1974, Lect. Notes Math. 480 (1975), 1-96 (In French.) DOI 10.1007/bfb0080190 | MR 0413238 | Zbl 0331.60025
[9] F. Flandoli: On the semigroup approach to stochastic evolution equations. Stochastic Anal. Appl. 10 (1992), 181-203. DOI 10.1080/07362999208809262 | MR 1154534 | Zbl 0762.60046
[10] M. J. Garrido-Atienza, B. Maslowski, J. Šnupárková: Semilinear stochastic equations with bilinear fractional noise. Discrete Contin. Dyn. Syst., Ser. B 21 (2016), 3075-3094. DOI 10.3934/dcdsb.2016088 | MR 3567802 | Zbl 1353.60059
[11] J. A. León, D. Nualart: Stochastic evolution equations with random generators. Ann. Probab. 26 (1998), 149-186. DOI 10.1214/aop/1022855415 | MR 1617045 | Zbl 0939.60066
[12] Y. S. Mishura: Quasi-linear stochastic differential equations with a fractional-Brownian component. Theory Probab. Math. Statist. 68 (2004), 103-115 (In English. Ukrainian original); translation from Teor. Ĭmovīr. Mat. Stat. 68 (2003), 95-106. DOI 10.1090/S0094-9000-04-00608-8 | MR 2000399 | Zbl 1050.60060
[13] Y. Mishura, G. Shevchenko: The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion. Stochastics 80 (2008), 489-511. DOI 10.1080/17442500802024892 | MR 2456334 | Zbl 1154.60046
[14] I. Nourdin, G. Peccati: Normal Approximations with Malliavin Calculus. From Stein's Method to Universality. Cambridge Tracts in Mathematics 192, Cambridge University Press, Cambridge (2012). DOI 10.1017/CBO9781139084659 | MR 2962301 | Zbl 1266.60001
[15] D. Nualart: The Malliavin Calculus and Related Topics. Probability and Its Applications, Springer, New York (1995). DOI 10.1007/978-1-4757-2437-0 | MR 1344217 | Zbl 0837.60050
[16] D. Nualart: Stochastic integration with respect to fractional Brownian motion and applications. Stochastic Models. Seventh Symposium on Probability and Stochastic Processes, Mexico City 2002 (J. M. Gonzáles-Barrios et al., eds.). Contemp. Math. 336, American Mathematical Society, Providence, 2003, pp. 3-39. DOI 10.1090/conm/336 | MR 2037156 | Zbl 1063.60080
[17] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44, Springer, New York (1983). DOI 10.1007/978-1-4612-5561-1 | MR 0710486 | Zbl 0516.47023
[18] V. Pérez-Abreu, C. Tudor: Multiple stochastic fractional integrals: a transfer principle for multiple stochastic fractional integrals. Bol. Soc. Mat. Mex., III. Ser. 8 (2002), 187-203. MR 1952159 | Zbl 1020.60050
[19] S. G. Samko, A. A. Kilbas, O. I. Marichev: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). MR 1347689 | Zbl 0818.26003
[20] J. Šnupárková: Stochastic bilinear equations with fractional Gaussian noise in Hilbert space. Acta Univ. Carol., Math. Phys. 51 (2010), 49-67. MR 2828153 | Zbl 1229.60073
[21] H. Tanabe: Equations of Evolution. Monographs and Studies in Mathematics 6, Pitman, London (1979). MR 0533824 | Zbl 0417.35003
Affiliations: Bohdan Maslowski, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: maslow@karlin.mff.cuni.cz; Jana Šnupárková, Department of Mathematics, Faculty of Chemical Engineering, University of Chemical Technology Prague, Studentská 6, 166 28 Praha 6, Czech Republic, e-mail: snuparkj@vscht.cz
