Global solutions and random dynamical systems for rough evolution equations

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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 and Continuous Dynamical Systems - B, 2020, 25 (7) : 2723-2748. doi: 10.3934/dcdsb.2020029

References:

[1]	L. Arnold, Random Dynamical Systems, Springer, Berlin Heidelberg, Germany, 2003.
[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.
[3]	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.
[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.
[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.
[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.
[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.
[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.
[9]	P. K. Friz and M. Hairer, A Course on Rough Paths, Springer, 2014. doi: 10.1007/978-3-319-08332-2.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	M. Gubinelli, Controlling rough paths, J. Func. Anal., 216 (2004), 86-140. doi: 10.1016/j.jfa.2004.01.002.
[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.
[21]	M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75. doi: 10.1214/08-AOP437.
[22]	M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504. doi: 10.1007/s00222-014-0505-4.
[23]	M. Hairer, Ergodicity of stochastic differential equations driven by fractional Brownian motion, Ann. Probab., 33 (2005), 703-758. doi: 10.1214/009117904000000892.
[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.
[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.
[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.
[27]	L. W. Kantorowitsch and G. P. Akilow, Funktionalanalysis in Normierten Räumen, Verlag Harri Deutsch, 1978.
[28]	H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.
[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.
[30]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995.
[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.
[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.
[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.
[34]	D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.
[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.
[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.
[37]	M. Scheutzow, On the perfection of crude cocycles, Random Comput. Dynam., 4 (1996), 235-255.
[38]	A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, 2010. doi: 10.1007/978-3-642-04631-5.
[39]	L. C. Young, An integration of Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282. doi: 10.1007/BF02401743.
[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.

show all references

References:

[1]	L. Arnold, Random Dynamical Systems, Springer, Berlin Heidelberg, Germany, 2003.
[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.
[3]	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.
[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.
[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.
[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.
[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.
[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.
[9]	P. K. Friz and M. Hairer, A Course on Rough Paths, Springer, 2014. doi: 10.1007/978-3-319-08332-2.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	M. Gubinelli, Controlling rough paths, J. Func. Anal., 216 (2004), 86-140. doi: 10.1016/j.jfa.2004.01.002.
[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.
[21]	M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75. doi: 10.1214/08-AOP437.
[22]	M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504. doi: 10.1007/s00222-014-0505-4.
[23]	M. Hairer, Ergodicity of stochastic differential equations driven by fractional Brownian motion, Ann. Probab., 33 (2005), 703-758. doi: 10.1214/009117904000000892.
[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.
[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.
[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.
[27]	L. W. Kantorowitsch and G. P. Akilow, Funktionalanalysis in Normierten Räumen, Verlag Harri Deutsch, 1978.
[28]	H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.
[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.
[30]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995.
[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.
[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.
[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.
[34]	D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.
[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.
[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.
[37]	M. Scheutzow, On the perfection of crude cocycles, Random Comput. Dynam., 4 (1996), 235-255.
[38]	A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, 2010. doi: 10.1007/978-3-642-04631-5.
[39]	L. C. Young, An integration of Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282. doi: 10.1007/BF02401743.
[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.

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