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July  2020, 25(7): 2723-2748. doi: 10.3934/dcdsb.2020029

Global solutions and random dynamical systems for rough evolution equations

Robert Hesse 1, and  Alexandra Neamţu 2,,

1. 

Friedrich Schiller University Jena, Enst-Abbe-Platz 2, 07743, Jena, Germany
2. 

Technical University of Munich, Boltzmannstr. 3, 85748, Garching bei München, Germany

* Corresponding author: Alexandra Neamţu

Received  April 2019 Published  July 2020 Early access  April 2020

Fund Project: The authors are grateful to M. J. Garrido-Atienza and B. Schmalfuß for helpful comments. The authors also thank the referee for the valuable suggestions. AN acknowledges support by a DFG grant in the D-A-CH framework (KU 3333/2-1)

We consider infinite-dimensional parabolic rough evolution equations. Using regularizing properties of analytic semigroups we prove global-in-time existence of solutions and investigate random dynamical systems for such equations.

Keywords: Stochastic evolution equations, random dynamical systems, rough paths theory, fractional Brownian motion.
Mathematics Subject Classification: 60H15, 60H05, 60G22, 37L55.
Citation: Robert Hesse, Alexandra Neamţu. Global solutions and random dynamical systems for rough evolution equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2723-2748. doi: 10.3934/dcdsb.2020029
References:
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L. Arnold, Random Dynamical Systems, Springer, Berlin Heidelberg, Germany, 2003. Google Scholar

[2]

I. Bailleul, S. Riedel and M. Scheutzow, Random dynamical system, rough paths and rough flows, J. Differential Equat., 262 (2017), 5792-5823.  doi: 10.1016/j.jde.2017.02.014.  Google Scholar

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L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory. Relat. Fields, 122 (2002), 108-140.  doi: 10.1007/s004400100158.  Google Scholar

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A. Deya, M. Gubinelli and S. Tindel, Non-linear rough heat equations, Probab. Theory Related Fields, 153 (2012), 97-147.  doi: 10.1007/s00440-011-0341-z.  Google Scholar

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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), 518-550.  doi: 10.1214/10-AIHP392.  Google Scholar

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

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H. Gao, M. J. Garrido-Atienza and 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.  Google Scholar

[12]

M. J. Garrido-Atienza and B. Schmalfuß, Local stability of differential equations driven by Hölder-continuous paths with Hölder index in (1/3, 1/2), SIAM J. Appl. Dyn. Syst., 17 (2018), 2352-2380.  doi: 10.1137/17M1160999.  Google Scholar

[13]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic partial differential equations driven by fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473-493.  doi: 10.3934/dcdsb.2010.14.473.  Google Scholar

[14]

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

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M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in(1/3, 1/2]$, Discrete Cont. Dyn-B., 20 (2015), 2553-2581.  doi: 10.3934/dcdsb.2015.20.2553.  Google Scholar

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M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Lévy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces, Continuous and Distributed Systems II, 30 (2015), 167-188.  doi: 10.1007/978-3-319-19075-4_10.  Google Scholar

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

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M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs, Potential Anal., 25 (2006), 307-326.  doi: 10.1007/s11118-006-9013-5.  Google Scholar

[21]

M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75.  doi: 10.1214/08-AOP437.  Google Scholar

[22]

M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504.  doi: 10.1007/s00222-014-0505-4.  Google Scholar

[23]

M. Hairer, Ergodicity of stochastic differential equations driven by fractional Brownian motion, Ann. Probab., 33 (2005), 703-758.  doi: 10.1214/009117904000000892.  Google Scholar

[24]

R. Hesse and A. Neamtu, Local mild solutions for rough stochastic partial differential equations, J. Differential Equat., 267 (2019), 6480-6538.  doi: 10.1016/j.jde.2019.06.026.  Google Scholar

[25]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer. Math. Soc., 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.  Google Scholar

[26]

P. Imkeller and C. Lederer, The cohomology of stochastic and random differential equations and local linearization of stochastic flows, Stoch. Dyn., 2 (2002), 131-159.  doi: 10.1142/S021949370200039X.  Google Scholar

[27]

L. W. Kantorowitsch and G. P. Akilow, Funktionalanalysis in Normierten Räumen, Verlag Harri Deutsch, 1978.  Google Scholar

[28]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.  Google Scholar

[29]

C. Lederer, Konjugation Stochastischer Und Zufälliger Stationärer Differentialgleichungen Und Eine Version Des Lokalen Satzes von Hartman-Grobman Für Stochastische Differentialgleichungen, PhD Thesis. HU Berlin, 2001. Google Scholar

[30]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995.  Google Scholar

[31]

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.  Google Scholar

[32]

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.  Google Scholar

[33]

S. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Memoirs of the AIMS, 196 (2008), vi+105 pp. doi: 10.1090/memo/0917.  Google Scholar

[34]

D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.   Google Scholar

[35]

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

[36]

R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer, 2002. doi: 10.1007/978-1-4471-3903-4.  Google Scholar

[37]

M. Scheutzow, On the perfection of crude cocycles, Random Comput. Dynam., 4 (1996), 235-255.   Google Scholar

[38]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

[39]

L. C. Young, An integration of Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282.  doi: 10.1007/BF02401743.  Google Scholar

[40]

M. Zähle, Integration with respect to fractal functions and stochastic calculus Ⅰ, Probab. Theory Related Fields, 111 (1998), 333-374.  doi: 10.1007/s004400050171.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer, Berlin Heidelberg, Germany, 2003. Google Scholar

[2]

I. Bailleul, S. Riedel and M. Scheutzow, Random dynamical system, rough paths and rough flows, J. Differential Equat., 262 (2017), 5792-5823.  doi: 10.1016/j.jde.2017.02.014.  Google Scholar

[3]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[4]

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

[5]

L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory. Relat. Fields, 122 (2002), 108-140.  doi: 10.1007/s004400100158.  Google Scholar

[6]

A. Deya, M. Gubinelli and S. Tindel, Non-linear rough heat equations, Probab. Theory Related Fields, 153 (2012), 97-147.  doi: 10.1007/s00440-011-0341-z.  Google Scholar

[7]

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), 518-550.  doi: 10.1214/10-AIHP392.  Google Scholar

[8]

B. Fehrmann and B. Gess, Well-posedness of stochastic porous media equations with nonlinear, conservative noise, Arch. Ration. Mech. Anal., 233 (2019), 249-322.  doi: 10.1007/s00205-019-01357-w.  Google Scholar

[9]

P. K. Friz and M. Hairer, A Course on Rough Paths, Springer, 2014. doi: 10.1007/978-3-319-08332-2.  Google Scholar

[10]

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

[11]

H. Gao, M. J. Garrido-Atienza and 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.  Google Scholar

[12]

M. J. Garrido-Atienza and B. Schmalfuß, Local stability of differential equations driven by Hölder-continuous paths with Hölder index in (1/3, 1/2), SIAM J. Appl. Dyn. Syst., 17 (2018), 2352-2380.  doi: 10.1137/17M1160999.  Google Scholar

[13]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic partial differential equations driven by fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473-493.  doi: 10.3934/dcdsb.2010.14.473.  Google Scholar

[14]

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

[15]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in(1/3, 1/2]$, Discrete Cont. Dyn-B., 20 (2015), 2553-2581.  doi: 10.3934/dcdsb.2015.20.2553.  Google Scholar

[16]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Lévy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces, Continuous and Distributed Systems II, 30 (2015), 167-188.  doi: 10.1007/978-3-319-19075-4_10.  Google Scholar

[17]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parametes $H\in(1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.  Google Scholar

[18]

M. Ghani Varzaneh, S. Riedel and M. Scheutzow, A dynamical theory for singular stochastic delay differential equations with a multiplicative ergodic theorem on fields of Banach spaces, arXiv: 1903.01172, 2019, 1–47. Google Scholar

[19]

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

[20]

M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs, Potential Anal., 25 (2006), 307-326.  doi: 10.1007/s11118-006-9013-5.  Google Scholar

[21]

M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75.  doi: 10.1214/08-AOP437.  Google Scholar

[22]

M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504.  doi: 10.1007/s00222-014-0505-4.  Google Scholar

[23]

M. Hairer, Ergodicity of stochastic differential equations driven by fractional Brownian motion, Ann. Probab., 33 (2005), 703-758.  doi: 10.1214/009117904000000892.  Google Scholar

[24]

R. Hesse and A. Neamtu, Local mild solutions for rough stochastic partial differential equations, J. Differential Equat., 267 (2019), 6480-6538.  doi: 10.1016/j.jde.2019.06.026.  Google Scholar

[25]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer. Math. Soc., 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.  Google Scholar

[26]

P. Imkeller and C. Lederer, The cohomology of stochastic and random differential equations and local linearization of stochastic flows, Stoch. Dyn., 2 (2002), 131-159.  doi: 10.1142/S021949370200039X.  Google Scholar

[27]

L. W. Kantorowitsch and G. P. Akilow, Funktionalanalysis in Normierten Räumen, Verlag Harri Deutsch, 1978.  Google Scholar

[28]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.  Google Scholar

[29]

C. Lederer, Konjugation Stochastischer Und Zufälliger Stationärer Differentialgleichungen Und Eine Version Des Lokalen Satzes von Hartman-Grobman Für Stochastische Differentialgleichungen, PhD Thesis. HU Berlin, 2001. Google Scholar

[30]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995.  Google Scholar

[31]

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.  Google Scholar

[32]

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.  Google Scholar

[33]

S. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Memoirs of the AIMS, 196 (2008), vi+105 pp. doi: 10.1090/memo/0917.  Google Scholar

[34]

D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.   Google Scholar

[35]

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

[36]

R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer, 2002. doi: 10.1007/978-1-4471-3903-4.  Google Scholar

[37]

M. Scheutzow, On the perfection of crude cocycles, Random Comput. Dynam., 4 (1996), 235-255.   Google Scholar

[38]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

[39]

L. C. Young, An integration of Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282.  doi: 10.1007/BF02401743.  Google Scholar

[40]

M. Zähle, Integration with respect to fractal functions and stochastic calculus Ⅰ, Probab. Theory Related Fields, 111 (1998), 333-374.  doi: 10.1007/s004400050171.  Google Scholar

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