August  2021, 26(8): 4587-4612. doi: 10.3934/dcdsb.2020304

A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds

1. 

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

2. 

Sharif University of Technology, Azadi Ave, Tehran, Iran

3. 

Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Germany

* Corresponding author: Sebastian Riedel

Received  March 2020 Published  August 2021 Early access  October 2020

Building on results obtained in [21], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [20].

Citation: Mazyar Ghani Varzaneh, Sebastian Riedel. A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4587-4612. doi: 10.3934/dcdsb.2020304
References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Second edition, Applied Mathematical Sciences, 75. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

I. Bailleul, Regularity of the Itô-Lyons map, Confluentes Math., 7 (2015), 3-11.  doi: 10.5802/cml.15.

[4]

P. Boxler, A stochastic version of center manifold theory, Probab. Theory Related Fields, 83 (1989), 509-545.  doi: 10.1007/BF01845701.

[5]

T. CaraballoJ. DuanK. Lu and B. Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.  doi: 10.1515/ans-2010-0102.

[6]

T. CaraballoJ. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2041-2061.  doi: 10.1098/rspa.2001.0819.

[7]

A. Carverhill, Flows of stochastic dynamical systems: Ergodic theory, Stochastics, 14 (1985), 273-317.  doi: 10.1080/17442508508833343.

[8]

M. D. Chekroun, H. Liu and S. Wang, Approximation of Stochastic Invariant Manifolds. Stochastic Manifolds for Nonlinear SPDEs. Ⅰ, SpringerBriefs in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-12496-4.

[9]

M. D. Chekroun, H. Liu and S. Wang, Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations. Stochastic Manifolds for Nonlinear SPDEs. Ⅱ, SpringerBriefs in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-12520-6.

[10]

X. ChenA. J. Roberts and J. Duan, Centre manifolds for stochastic evolution equations, J. Difference Equ. Appl., 21 (2015), 606-632.  doi: 10.1080/10236198.2015.1045889.

[11]

X. ChenA. J. Roberts and J. Duan, Centre manifolds for infinite dimensional random dynamical systems, Dyn. Syst., 34 (2019), 334-355.  doi: 10.1080/14689367.2018.1531972.

[12]

L. Coutin and A. Lejay, Sensitivity of rough differential equations: An approach through the omega lemma, J. Differential Equations, 264 (2018), 3899-3917.  doi: 10.1016/j.jde.2017.11.031.

[13]

A. Du and J. Duan, Invariant manifold reduction for stochastic dynamical systems, Dynam. Systems Appl., 16 (2007), 681-696. 

[14]

J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.

[15]

J. DuanK. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.

[16]

J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Insights, Elsevier, Amsterdam, 2014.

[17]

P. K. Friz and M. Hairer, A Course on Rough Paths. With an Introduction to Regularity Structures, Universitext, Springer, Berlin, 2014. doi: 10.1007/978-3-319-08332-2.

[18] P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511845079.
[19]

M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations, 248 (2010), 1637-1667.  doi: 10.1016/j.jde.2009.11.006.

[20]

M. G. Varzaneh and S. Riedel, Oseledets splitting and invariant manifolds on fields of Banach spaces, 2019, arXiv: 1912.07985.

[21]

M. G. Varzaneh, S. Riedel and M. Scheutzow, A dynamical theory for singular stochastic delay differential equations Ⅰ: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces, 2019, arXiv: 1903.01172v3.

[22]

M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140.  doi: 10.1016/j.jfa.2004.01.002.

[23]

E. Knobloch and K. A. Wiesenfeld, Bifurcations in fluctuating systems: The center-manifold approach, J. Statist. Phys., 33 (1983), 611-637.  doi: 10.1007/BF01018837.

[24]

C. Kuehn and A. Neamţu, Rough center manifolds, 2018, arXiv: 1811.10037.

[25]

K. LuA. Neamţu and B. Schmalfuss, On the Oseledets-splitting for infinite-dimensional random dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1219-1242.  doi: 10.3934/dcdsb.2018149.

[26]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  doi: 10.4171/RMI/240.

[27]

S. E. A. Mohammed, Nonlinear flows of stochastic linear delay equations, Stochastics, 17 (1986), 207-213.  doi: 10.1080/17442508608833390.

[28]

S. E. A. Mohammed and M. K. R. Scheutzow, Lyapunov exponents and stationary solutions for affine stochastic delay equations, Stochastics Stochastics Rep., 29 (1990), 259-283.  doi: 10.1080/17442509008833617.

[29]

S.-E. A. Mohammed and M. K. R. Scheutzow, Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. Ⅰ. The multiplicative ergodic theory, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 69-105. 

[30]

S.-E. A. Mohammed and M. K. R. Scheutzow, Lyapunov exponents of linear stochastic functional-differential equations. Ⅱ. Examples and case studies, Ann. Probab., 25 (1997), 1210-1240.  doi: 10.1214/aop/1024404511.

[31]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, Ann. Probab., 27 (1999), 615-652.  doi: 10.1214/aop/1022677380.

[32]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory. Ⅰ. Existence of the semiflow, J. Funct. Anal., 205 (2003), 271-305.  doi: 10.1016/j.jfa.2002.04.001.

[33]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory. Ⅱ. The local stable manifold theorem, J. Funct. Anal., 206 (2004), 253-306.  doi: 10.1016/j.jfa.2003.06.002.

[34]

S. Mohammed and T. Zhang, Dynamics of stochastic 2D Navier-Stokes equations, J. Funct. Anal., 258 (2010), 3543-3591.  doi: 10.1016/j.jfa.2009.11.007.

[35]

S.-E. A. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., 196 (2008), vi+105 pp. doi: 10.1090/memo/0917.

[36]

A. Neamţu, Random invariant manifolds for ill-posed stochastic evolution equations, Stochastics and Dynamics, 20 (2020), 2050013, 31 pp. doi: 10.1142/S0219493720500136.

[37]

A. NeuenkirchI. Nourdin and S. Tindel, Delay equations driven by rough paths, Electron. J. Probab., 13 (2008), 2031-2068.  doi: 10.1214/EJP.v13-575.

[38]

S. Riedel and M. Scheutzow, Rough differential equations with unbounded drift term, J. Differential Equations, 262 (2017), 283-312.  doi: 10.1016/j.jde.2016.09.021.

[39]

T. Wanner, Linearization of random dynamical systems, Dynamics Reported, Dynam. Report. Expositions Dynam. Systems (N.S.), Springer, Berlin, 4 (1995), 203-269. 

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Second edition, Applied Mathematical Sciences, 75. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

I. Bailleul, Regularity of the Itô-Lyons map, Confluentes Math., 7 (2015), 3-11.  doi: 10.5802/cml.15.

[4]

P. Boxler, A stochastic version of center manifold theory, Probab. Theory Related Fields, 83 (1989), 509-545.  doi: 10.1007/BF01845701.

[5]

T. CaraballoJ. DuanK. Lu and B. Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.  doi: 10.1515/ans-2010-0102.

[6]

T. CaraballoJ. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2041-2061.  doi: 10.1098/rspa.2001.0819.

[7]

A. Carverhill, Flows of stochastic dynamical systems: Ergodic theory, Stochastics, 14 (1985), 273-317.  doi: 10.1080/17442508508833343.

[8]

M. D. Chekroun, H. Liu and S. Wang, Approximation of Stochastic Invariant Manifolds. Stochastic Manifolds for Nonlinear SPDEs. Ⅰ, SpringerBriefs in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-12496-4.

[9]

M. D. Chekroun, H. Liu and S. Wang, Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations. Stochastic Manifolds for Nonlinear SPDEs. Ⅱ, SpringerBriefs in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-12520-6.

[10]

X. ChenA. J. Roberts and J. Duan, Centre manifolds for stochastic evolution equations, J. Difference Equ. Appl., 21 (2015), 606-632.  doi: 10.1080/10236198.2015.1045889.

[11]

X. ChenA. J. Roberts and J. Duan, Centre manifolds for infinite dimensional random dynamical systems, Dyn. Syst., 34 (2019), 334-355.  doi: 10.1080/14689367.2018.1531972.

[12]

L. Coutin and A. Lejay, Sensitivity of rough differential equations: An approach through the omega lemma, J. Differential Equations, 264 (2018), 3899-3917.  doi: 10.1016/j.jde.2017.11.031.

[13]

A. Du and J. Duan, Invariant manifold reduction for stochastic dynamical systems, Dynam. Systems Appl., 16 (2007), 681-696. 

[14]

J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.

[15]

J. DuanK. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.

[16]

J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Insights, Elsevier, Amsterdam, 2014.

[17]

P. K. Friz and M. Hairer, A Course on Rough Paths. With an Introduction to Regularity Structures, Universitext, Springer, Berlin, 2014. doi: 10.1007/978-3-319-08332-2.

[18] P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511845079.
[19]

M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations, 248 (2010), 1637-1667.  doi: 10.1016/j.jde.2009.11.006.

[20]

M. G. Varzaneh and S. Riedel, Oseledets splitting and invariant manifolds on fields of Banach spaces, 2019, arXiv: 1912.07985.

[21]

M. G. Varzaneh, S. Riedel and M. Scheutzow, A dynamical theory for singular stochastic delay differential equations Ⅰ: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces, 2019, arXiv: 1903.01172v3.

[22]

M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140.  doi: 10.1016/j.jfa.2004.01.002.

[23]

E. Knobloch and K. A. Wiesenfeld, Bifurcations in fluctuating systems: The center-manifold approach, J. Statist. Phys., 33 (1983), 611-637.  doi: 10.1007/BF01018837.

[24]

C. Kuehn and A. Neamţu, Rough center manifolds, 2018, arXiv: 1811.10037.

[25]

K. LuA. Neamţu and B. Schmalfuss, On the Oseledets-splitting for infinite-dimensional random dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1219-1242.  doi: 10.3934/dcdsb.2018149.

[26]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  doi: 10.4171/RMI/240.

[27]

S. E. A. Mohammed, Nonlinear flows of stochastic linear delay equations, Stochastics, 17 (1986), 207-213.  doi: 10.1080/17442508608833390.

[28]

S. E. A. Mohammed and M. K. R. Scheutzow, Lyapunov exponents and stationary solutions for affine stochastic delay equations, Stochastics Stochastics Rep., 29 (1990), 259-283.  doi: 10.1080/17442509008833617.

[29]

S.-E. A. Mohammed and M. K. R. Scheutzow, Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. Ⅰ. The multiplicative ergodic theory, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 69-105. 

[30]

S.-E. A. Mohammed and M. K. R. Scheutzow, Lyapunov exponents of linear stochastic functional-differential equations. Ⅱ. Examples and case studies, Ann. Probab., 25 (1997), 1210-1240.  doi: 10.1214/aop/1024404511.

[31]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, Ann. Probab., 27 (1999), 615-652.  doi: 10.1214/aop/1022677380.

[32]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory. Ⅰ. Existence of the semiflow, J. Funct. Anal., 205 (2003), 271-305.  doi: 10.1016/j.jfa.2002.04.001.

[33]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory. Ⅱ. The local stable manifold theorem, J. Funct. Anal., 206 (2004), 253-306.  doi: 10.1016/j.jfa.2003.06.002.

[34]

S. Mohammed and T. Zhang, Dynamics of stochastic 2D Navier-Stokes equations, J. Funct. Anal., 258 (2010), 3543-3591.  doi: 10.1016/j.jfa.2009.11.007.

[35]

S.-E. A. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., 196 (2008), vi+105 pp. doi: 10.1090/memo/0917.

[36]

A. Neamţu, Random invariant manifolds for ill-posed stochastic evolution equations, Stochastics and Dynamics, 20 (2020), 2050013, 31 pp. doi: 10.1142/S0219493720500136.

[37]

A. NeuenkirchI. Nourdin and S. Tindel, Delay equations driven by rough paths, Electron. J. Probab., 13 (2008), 2031-2068.  doi: 10.1214/EJP.v13-575.

[38]

S. Riedel and M. Scheutzow, Rough differential equations with unbounded drift term, J. Differential Equations, 262 (2017), 283-312.  doi: 10.1016/j.jde.2016.09.021.

[39]

T. Wanner, Linearization of random dynamical systems, Dynamics Reported, Dynam. Report. Expositions Dynam. Systems (N.S.), Springer, Berlin, 4 (1995), 203-269. 

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