January  2014, 34(1): 79-98. doi: 10.3934/dcds.2014.34.79

Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems

1. 

School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023

2. 

Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

3. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain

4. 

Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 07737 Jena

Received  November 2012 Revised  March 2013 Published  June 2013

This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of Hölder continuous functions with values in a Hilbert space $V$. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the Hölder continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing $B^H$ as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion.
Citation: Yong Chen, Hongjun Gao, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 79-98. doi: 10.3934/dcds.2014.34.79
References:
[1]

M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", Dover Publications, (1966).   Google Scholar

[2]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar

[3]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equation without uniqueness of solutions,, Discrete and Continuous Dynamical Systems, 14 (2010), 439.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[4]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions,", Lecture Notes in Mathematics, (1977).   Google Scholar

[5]

Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Random attractors for SPDEs driven by a fractional Brownian motion,, in preperation., ().   Google Scholar

[6]

P. Friz and N. Victoir, "Multidimensional Stochastic Processes as Rough Paths. Theory and Applications,", Cambridge Studies of Advanced Mathematics, (2010).   Google Scholar

[7]

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 Systems, 14 (2010), 473.  doi: 10.3934/dcdsb.2010.14.473.  Google Scholar

[8]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by a fractional Brownian motion with Hurst parameter in (1/3,1/2],, in preparation., ().   Google Scholar

[9]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Pathwise solutions of stochastic partial differential equations driven by a fractional Brownian motion with Hurst parameter in (1/3,1/2],, arXiv1205.6735., ().   Google Scholar

[10]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Compensated fractional derivatives and stochastic evolution equations,, Comptes Rendus Mathématique, 350 (2012), 1037.  doi: 10.1016/j.crma.2012.11.007.  Google Scholar

[11]

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.  doi: 10.1142/S0218127410027349.  Google Scholar

[12]

M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion,, Journal of Dynamics and Differential Equations, 23 (2011), 671.  doi: 10.1007/s10884-011-9222-5.  Google Scholar

[13]

W. Grecksch and V. V. Anh, A parabolic stochastic differential equation with fractional Brownian motion input,, Statist. Probab. Lett., 41 (1999), 337.  doi: 10.1016/S0167-7152(98)00147-3.  Google Scholar

[14]

M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs,, Potential Anal., 25 (2006), 307.  doi: 10.1007/s11118-006-9013-5.  Google Scholar

[15]

H. Kunita, "Stochastic Flows and Stochastic Differential Equations,", Cambridge Studies in Advanced Mathematics, 24 (1990).   Google Scholar

[16]

T. Lyons and Z. Qian, "System Control and Rough Paths,", Oxford Mathematical Monographs, (2002).  doi: 10.1093/acprof:oso/9780198506485.001.0001.  Google Scholar

[17]

B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion,, J. Funct. Anal., 202 (2003), 277.  doi: 10.1016/S0022-1236(02)00065-4.  Google Scholar

[18]

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.  doi: 10.1081/SAP-200029498.  Google Scholar

[19]

D. Nualart and A. Răçcanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55.   Google Scholar

[20]

S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives: Theory and Applications,", Gordon and Breach Science Publishers, (1993).   Google Scholar

[21]

S. Tindel, C. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion,, Probability Theory and Related Fields, 127 (2003), 186.  doi: 10.1007/s00440-003-0282-2.  Google Scholar

[22]

M. Zähle, Integration with respect to fractal functions and stochastic calculus. I,, Probab. Theory Related Fields, 111 (1998), 333.  doi: 10.1007/s004400050171.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", Dover Publications, (1966).   Google Scholar

[2]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar

[3]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equation without uniqueness of solutions,, Discrete and Continuous Dynamical Systems, 14 (2010), 439.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[4]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions,", Lecture Notes in Mathematics, (1977).   Google Scholar

[5]

Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Random attractors for SPDEs driven by a fractional Brownian motion,, in preperation., ().   Google Scholar

[6]

P. Friz and N. Victoir, "Multidimensional Stochastic Processes as Rough Paths. Theory and Applications,", Cambridge Studies of Advanced Mathematics, (2010).   Google Scholar

[7]

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 Systems, 14 (2010), 473.  doi: 10.3934/dcdsb.2010.14.473.  Google Scholar

[8]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by a fractional Brownian motion with Hurst parameter in (1/3,1/2],, in preparation., ().   Google Scholar

[9]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Pathwise solutions of stochastic partial differential equations driven by a fractional Brownian motion with Hurst parameter in (1/3,1/2],, arXiv1205.6735., ().   Google Scholar

[10]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Compensated fractional derivatives and stochastic evolution equations,, Comptes Rendus Mathématique, 350 (2012), 1037.  doi: 10.1016/j.crma.2012.11.007.  Google Scholar

[11]

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.  doi: 10.1142/S0218127410027349.  Google Scholar

[12]

M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion,, Journal of Dynamics and Differential Equations, 23 (2011), 671.  doi: 10.1007/s10884-011-9222-5.  Google Scholar

[13]

W. Grecksch and V. V. Anh, A parabolic stochastic differential equation with fractional Brownian motion input,, Statist. Probab. Lett., 41 (1999), 337.  doi: 10.1016/S0167-7152(98)00147-3.  Google Scholar

[14]

M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs,, Potential Anal., 25 (2006), 307.  doi: 10.1007/s11118-006-9013-5.  Google Scholar

[15]

H. Kunita, "Stochastic Flows and Stochastic Differential Equations,", Cambridge Studies in Advanced Mathematics, 24 (1990).   Google Scholar

[16]

T. Lyons and Z. Qian, "System Control and Rough Paths,", Oxford Mathematical Monographs, (2002).  doi: 10.1093/acprof:oso/9780198506485.001.0001.  Google Scholar

[17]

B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion,, J. Funct. Anal., 202 (2003), 277.  doi: 10.1016/S0022-1236(02)00065-4.  Google Scholar

[18]

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.  doi: 10.1081/SAP-200029498.  Google Scholar

[19]

D. Nualart and A. Răçcanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55.   Google Scholar

[20]

S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives: Theory and Applications,", Gordon and Breach Science Publishers, (1993).   Google Scholar

[21]

S. Tindel, C. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion,, Probability Theory and Related Fields, 127 (2003), 186.  doi: 10.1007/s00440-003-0282-2.  Google Scholar

[22]

M. Zähle, Integration with respect to fractal functions and stochastic calculus. I,, Probab. Theory Related Fields, 111 (1998), 333.  doi: 10.1007/s004400050171.  Google Scholar

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