American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020304

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

Mazyar Ghani Varzaneh 1,2, and  Sebastian Riedel 1,3,,

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  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].

Keywords: Random dynamical systems, rough paths, stable and unstable manifolds, stochastic delay differential equations.
Mathematics Subject Classification: Primary:60H20, 60L20;Secondary:34K19, 34K50, 37D10, 37H15.
Citation: Mazyar Ghani Varzaneh, Sebastian Riedel. A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[2]

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

[3]

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

[4]

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

[5]

T. Caraballo, J. Duan, K. 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.  Google Scholar

[6]

T. Caraballo, J. 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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

X. Chen, A. 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.  Google Scholar

[11]

X. Chen, A. 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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

J. Duan, K. 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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar
[19]

M. J. Garrido-Atienza, K. 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.  Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

K. Lu, A. 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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[39]

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

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.  Google Scholar

[2]

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

[3]

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

[4]

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

[5]

T. Caraballo, J. Duan, K. 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.  Google Scholar

[6]

T. Caraballo, J. 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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

X. Chen, A. 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.  Google Scholar

[11]

X. Chen, A. 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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

J. Duan, K. 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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar
[19]

M. J. Garrido-Atienza, K. 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.  Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

K. Lu, A. 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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[39]

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

[1]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192
[2]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[3]

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, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[4]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022
[5]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[6]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[7]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027
[8]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[9]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[10]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[11]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066
[12]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[13]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066
[14]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004
[15]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[16]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[17]

Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207
[18]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[19]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[20]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

2019 Impact Factor: 1.27

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences