American Institute of Mathematical Sciences

November  2016, 21(9): 3075-3094. doi: 10.3934/dcdsb.2016088

Semilinear stochastic equations with bilinear fractional noise

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico , Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla 2 Charles University in Prague, Faculty of Mathematics and Physics, Sokolovska 83, Prague 8, Czech Republic, Czech Republic

Received  February 2016 Revised  March 2016 Published  October 2016

In the paper, we study existence and uniqueness of solutions to semilinear stochastic evolution systems, driven by a fractional Brownian motion with bilinear noise term, and the long time behavior of solutions to such equations. For this purpose, we study at first the random evolution operator defined by the corresponding bilinear equation which is later used to define the mild solution of the semilinear equation. The mild solution is also shown to be weak in the PDE sense. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. We show that the solution generates a random dynamical system that, under appropriate stability and compactness conditions, possesses a random attractor.
Citation: María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
References:
 [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [2] H. Bessaih, M. J. Garrido-Atienza and B. Schmalfuß, Stochastic Shell Models driven by a multiplicative fractional Brownian motion, Physica D: Nonlinear Phenomena, 320 (2016), 38-56. doi: 10.1016/j.physd.2016.01.008. [3] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin, 1977. [4] Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 79-98. doi: 10.3934/dcds.2014.34.79. [5] J. W. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Math. J., 54 (2004), 991-1013. doi: 10.1007/s10587-004-6447-z. [6] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. [7] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochastic Process. Appl., 115 (2005), 1357-1383. doi: 10.1016/j.spa.2005.03.011. [8] F. Flandoli and B. Schmalfuß, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. [9] H. Gao, M. J. Garrido-Atienza, B. Schmalfuß, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309. doi: 10.1137/130930662. [10] M. J. Garrido-Atienza, P. Kloeden and A. Neuenkirch, Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion, Applied Mathematics and Optimization, 60 (2009), 151-172. doi: 10.1007/s00245-008-9062-9. [11] M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete and Continuous Dynamical System, Series B, 14 (2010), 473-493. doi: 10.3934/dcdsb.2010.14.473. [12] M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion, International Journal of Bifurcation and Chaos, 20 (2010), 2761-2782. doi: 10.1142/S0218127410027349. [13] M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion, Journal of Dynamics and Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5. [14] B. Gess, Random Attractors for Stochastic Porous Media Equations perturbed by space-time linear multiplicative noise, C.R. Acad. Sci. Paris, Ser. I, 350 (2012), 299-302. doi: 10.1016/j.crma.2012.02.004. [15] B. Gess, W. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, Journal of Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013. [16] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. [17] M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge Studies in Advanced Mathematics, 100, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511617997. [18] B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202 (2003), 277-305. doi: 10.1016/S0022-1236(02)00065-4. [19] B. Maslowski and B. Schmalfuß, Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stochastic Anal. Appl., 22 (2004), 1577-1607. doi: 10.1081/SAP-200029498. [20] B. Maslowski and J. Šnupárková, Stochastic equations with multiplicative fractional noise in Hilbert space, preprint, arXiv:1609.00582. [21] D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [23] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993. [24] B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in Int. Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), 1992, 185-192. [25] B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in Int. Conf. Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 684-689. [26] J. Šnupárková, Stochastic bilinear equations with fractional Gaussian noise in Hilbert space, Acta Univ. Carolin. Math. Phys., 51 (2010), 49-67. [27] M. Zähle, Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Relat. Fields, 111 (1998), 333-374. doi: 10.1007/s004400050171. [28] M. Zähle, Integration with respect to fractal functions and stochastic calculus II, Math. Nachr., 225 (2001), 145-183. doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0.

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References:
 [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [2] H. Bessaih, M. J. Garrido-Atienza and B. Schmalfuß, Stochastic Shell Models driven by a multiplicative fractional Brownian motion, Physica D: Nonlinear Phenomena, 320 (2016), 38-56. doi: 10.1016/j.physd.2016.01.008. [3] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin, 1977. [4] Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 79-98. doi: 10.3934/dcds.2014.34.79. [5] J. W. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Math. J., 54 (2004), 991-1013. doi: 10.1007/s10587-004-6447-z. [6] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. [7] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochastic Process. Appl., 115 (2005), 1357-1383. doi: 10.1016/j.spa.2005.03.011. [8] F. Flandoli and B. Schmalfuß, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. [9] H. Gao, M. J. Garrido-Atienza, B. Schmalfuß, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309. doi: 10.1137/130930662. [10] M. J. Garrido-Atienza, P. Kloeden and A. Neuenkirch, Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion, Applied Mathematics and Optimization, 60 (2009), 151-172. doi: 10.1007/s00245-008-9062-9. [11] M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete and Continuous Dynamical System, Series B, 14 (2010), 473-493. doi: 10.3934/dcdsb.2010.14.473. [12] M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion, International Journal of Bifurcation and Chaos, 20 (2010), 2761-2782. doi: 10.1142/S0218127410027349. [13] M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion, Journal of Dynamics and Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5. [14] B. Gess, Random Attractors for Stochastic Porous Media Equations perturbed by space-time linear multiplicative noise, C.R. Acad. Sci. Paris, Ser. I, 350 (2012), 299-302. doi: 10.1016/j.crma.2012.02.004. [15] B. Gess, W. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, Journal of Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013. [16] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. [17] M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge Studies in Advanced Mathematics, 100, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511617997. [18] B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202 (2003), 277-305. doi: 10.1016/S0022-1236(02)00065-4. [19] B. Maslowski and B. Schmalfuß, Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stochastic Anal. Appl., 22 (2004), 1577-1607. doi: 10.1081/SAP-200029498. [20] B. Maslowski and J. Šnupárková, Stochastic equations with multiplicative fractional noise in Hilbert space, preprint, arXiv:1609.00582. [21] D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [23] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993. [24] B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in Int. Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), 1992, 185-192. [25] B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in Int. Conf. Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 684-689. [26] J. Šnupárková, Stochastic bilinear equations with fractional Gaussian noise in Hilbert space, Acta Univ. Carolin. Math. Phys., 51 (2010), 49-67. [27] M. Zähle, Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Relat. Fields, 111 (1998), 333-374. doi: 10.1007/s004400050171. [28] M. Zähle, Integration with respect to fractal functions and stochastic calculus II, Math. Nachr., 225 (2001), 145-183. doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0.
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