• Previous Article
    Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one
  • DCDS-B Home
  • This Issue
  • Next Article
    On the homogenization of multicomponent transport
October  2015, 20(8): 2553-2581. doi: 10.3934/dcdsb.2015.20.2553

Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$

1. 

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

2. 

346 TMCB Brigham Young University, Provo, UT 84602

3. 

Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 77043, Jena, Germany

Received  November 2014 Revised  March 2015 Published  August 2015

In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting of a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which in particular includes white noise.
Citation: María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553
References:
[1]

M. Caruana and P. Friz, Partial differential equations driven by rough paths,, J. Differential Equations, 247 (2009), 140. doi: 10.1016/j.jde.2009.01.026. Google Scholar

[2]

M. Caruana, P. Friz and H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 27. doi: 10.1016/j.anihpc.2010.11.002. Google Scholar

[3]

Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems,, Discrete and Continuous Dynamical Systems, 34 (2014), 79. doi: 10.3934/dcds.2014.34.79. Google Scholar

[4]

A. Deya, A. Neuenkirch and S. Tindel, A Milstein-type scheme without Lévy area terms for SDES driven by fractional Brownian motion,, Ann. Inst. H. Poincaré Probab. Statist., 48 (2012), 518. doi: 10.1214/10-AIHP392. Google Scholar

[5]

A. Deya, M. Gubinelli and S. Tindel, Non-linear rough heat equations,, Probab. Theory Relat. Fields, 153 (2012), 97. doi: 10.1007/s00440-011-0341-z. Google Scholar

[6]

P. Friz and H. Oberhauser, On the splitting-up method for rough (partial) differential equations,, J. Differential Equations, 251 (2011), 316. doi: 10.1016/j.jde.2011.02.009. Google Scholar

[7]

P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications,, Cambridge Studies of Advanced Mathematics, (2010). doi: 10.1017/CBO9780511845079. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Lévy areas of Ornstein-Uhlenbeck processes in Hilbert spaces,, Studies in Systems, 30 (2015), 167. doi: 10.1007/978-3-319-19075-4_10. Google Scholar

[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. doi: 10.1142/S0218127410027349. Google Scholar

[13]

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

[14]

M. Gubinelli and S. Tindel, Rough Evolution Equations,, The Annals of Probability, 38 (2010), 1. doi: 10.1214/08-AOP437. Google Scholar

[15]

M. Hinz and M. Zähle, Gradient type noises II-Systems of stochastic partial differential equations,, Journal of Functional Analysis, 256 (2009), 3192. doi: 10.1016/j.jfa.2009.02.006. Google Scholar

[16]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus,, Trans. Amer. Math. Soc., 361 (2009), 2689. doi: 10.1090/S0002-9947-08-04631-X. Google Scholar

[17]

R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary theory,, Graduate Studies in Mathematics, (1997). Google Scholar

[18]

T. Lyons and Z. Qian, System control and rough paths,, Oxford Mathematical Monographs, (2002). doi: 10.1093/acprof:oso/9780198506485.001.0001. Google Scholar

[19]

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

[20]

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

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Applied Mathematical Series, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[22]

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

[23]

L. C. Young, An integration of Höder type, connected with Stieltjes integration,, Acta Math., 67 (1936), 251. doi: 10.1007/BF02401743. Google Scholar

[24]

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. Caruana and P. Friz, Partial differential equations driven by rough paths,, J. Differential Equations, 247 (2009), 140. doi: 10.1016/j.jde.2009.01.026. Google Scholar

[2]

M. Caruana, P. Friz and H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 27. doi: 10.1016/j.anihpc.2010.11.002. Google Scholar

[3]

Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems,, Discrete and Continuous Dynamical Systems, 34 (2014), 79. doi: 10.3934/dcds.2014.34.79. Google Scholar

[4]

A. Deya, A. Neuenkirch and S. Tindel, A Milstein-type scheme without Lévy area terms for SDES driven by fractional Brownian motion,, Ann. Inst. H. Poincaré Probab. Statist., 48 (2012), 518. doi: 10.1214/10-AIHP392. Google Scholar

[5]

A. Deya, M. Gubinelli and S. Tindel, Non-linear rough heat equations,, Probab. Theory Relat. Fields, 153 (2012), 97. doi: 10.1007/s00440-011-0341-z. Google Scholar

[6]

P. Friz and H. Oberhauser, On the splitting-up method for rough (partial) differential equations,, J. Differential Equations, 251 (2011), 316. doi: 10.1016/j.jde.2011.02.009. Google Scholar

[7]

P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications,, Cambridge Studies of Advanced Mathematics, (2010). doi: 10.1017/CBO9780511845079. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Lévy areas of Ornstein-Uhlenbeck processes in Hilbert spaces,, Studies in Systems, 30 (2015), 167. doi: 10.1007/978-3-319-19075-4_10. Google Scholar

[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. doi: 10.1142/S0218127410027349. Google Scholar

[13]

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

[14]

M. Gubinelli and S. Tindel, Rough Evolution Equations,, The Annals of Probability, 38 (2010), 1. doi: 10.1214/08-AOP437. Google Scholar

[15]

M. Hinz and M. Zähle, Gradient type noises II-Systems of stochastic partial differential equations,, Journal of Functional Analysis, 256 (2009), 3192. doi: 10.1016/j.jfa.2009.02.006. Google Scholar

[16]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus,, Trans. Amer. Math. Soc., 361 (2009), 2689. doi: 10.1090/S0002-9947-08-04631-X. Google Scholar

[17]

R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary theory,, Graduate Studies in Mathematics, (1997). Google Scholar

[18]

T. Lyons and Z. Qian, System control and rough paths,, Oxford Mathematical Monographs, (2002). doi: 10.1093/acprof:oso/9780198506485.001.0001. Google Scholar

[19]

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

[20]

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

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Applied Mathematical Series, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[22]

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

[23]

L. C. Young, An integration of Höder type, connected with Stieltjes integration,, Acta Math., 67 (1936), 251. doi: 10.1007/BF02401743. Google Scholar

[24]

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

[1]

Anna Karczewska, Carlos Lizama. On stochastic fractional Volterra equations in Hilbert space. Conference Publications, 2007, 2007 (Special) : 541-550. doi: 10.3934/proc.2007.2007.541

[2]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255

[3]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197

[4]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257

[5]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199

[6]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

[7]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[8]

Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483

[9]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084

[10]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019213

[11]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157

[12]

Mahmoud M. El-Borai. On some fractional differential equations in the Hilbert space. Conference Publications, 2005, 2005 (Special) : 233-240. doi: 10.3934/proc.2005.2005.233

[13]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

[14]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[15]

Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047

[16]

Giuseppe Da Prato, Franco Flandoli. Some results for pathwise uniqueness in Hilbert spaces. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1789-1797. doi: 10.3934/cpaa.2014.13.1789

[17]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817

[18]

S. Kanagawa, K. Inoue, A. Arimoto, Y. Saisho. Mean square approximation of multi dimensional reflecting fractional Brownian motion via penalty method. Conference Publications, 2005, 2005 (Special) : 463-475. doi: 10.3934/proc.2005.2005.463

[19]

Stefan Koch, Andreas Neuenkirch. The Mandelbrot-van Ness fractional Brownian motion is infinitely differentiable with respect to its Hurst parameter. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3865-3880. doi: 10.3934/dcdsb.2018334

[20]

Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (3)

[Back to Top]