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Probabilistic continuity of a pullback random attractor in time-sample
Global solutions and random dynamical systems for rough evolution equations
1. | Friedrich Schiller University Jena, Enst-Abbe-Platz 2, 07743, Jena, Germany |
2. | Technical University of Munich, Boltzmannstr. 3, 85748, Garching bei München, Germany |
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.
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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