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On the homogenization of multicomponent transport
Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$
1. | Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain |
2. | 346 TMCB Brigham Young University, Provo, UT 84602 |
3. | Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 77043, Jena, Germany |
References:
[1] |
M. Caruana and P. Friz, Partial differential equations driven by rough paths,, J. Differential Equations, 247 (2009), 140.
doi: 10.1016/j.jde.2009.01.026. |
[2] |
M. Caruana, P. Friz and H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 27.
doi: 10.1016/j.anihpc.2010.11.002. |
[3] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems,, Discrete and Continuous Dynamical Systems, 34 (2014), 79.
doi: 10.3934/dcds.2014.34.79. |
[4] |
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.
doi: 10.1214/10-AIHP392. |
[5] |
A. Deya, M. Gubinelli and S. Tindel, Non-linear rough heat equations,, Probab. Theory Relat. Fields, 153 (2012), 97.
doi: 10.1007/s00440-011-0341-z. |
[6] |
P. Friz and H. Oberhauser, On the splitting-up method for rough (partial) differential equations,, J. Differential Equations, 251 (2011), 316.
doi: 10.1016/j.jde.2011.02.009. |
[7] |
P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications,, Cambridge Studies of Advanced Mathematics, (2010).
doi: 10.1017/CBO9780511845079. |
[8] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion,, Discrete and Continuous Dynamical Systems, 14 (2010), 473.
doi: 10.3934/dcdsb.2010.14.473. |
[9] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Compensated fractional derivatives and stochastic evolution equations,, Comptes Rendus Mathématique, 350 (2012), 1037.
doi: 10.1016/j.crma.2012.11.007. |
[10] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameter $H\in (1/3,1/2]$,, , (). Google Scholar |
[11] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Lévy areas of Ornstein-Uhlenbeck processes in Hilbert spaces,, Studies in Systems, 30 (2015), 167.
doi: 10.1007/978-3-319-19075-4_10. |
[12] |
M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion,, International Journal of Bifurcation and Chaos, 20 (2010), 2761.
doi: 10.1142/S0218127410027349. |
[13] |
M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs,, Potential Anal., 25 (2006), 307.
doi: 10.1007/s11118-006-9013-5. |
[14] |
M. Gubinelli and S. Tindel, Rough Evolution Equations,, The Annals of Probability, 38 (2010), 1.
doi: 10.1214/08-AOP437. |
[15] |
M. Hinz and M. Zähle, Gradient type noises II-Systems of stochastic partial differential equations,, Journal of Functional Analysis, 256 (2009), 3192.
doi: 10.1016/j.jfa.2009.02.006. |
[16] |
Y. Hu and D. Nualart, Rough path analysis via fractional calculus,, Trans. Amer. Math. Soc., 361 (2009), 2689.
doi: 10.1090/S0002-9947-08-04631-X. |
[17] |
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary theory,, Graduate Studies in Mathematics, (1997).
|
[18] |
T. Lyons and Z. Qian, System control and rough paths,, Oxford Mathematical Monographs, (2002).
doi: 10.1093/acprof:oso/9780198506485.001.0001. |
[19] |
B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion,, J. Funct. Anal., 202 (2003), 277.
doi: 10.1016/S0022-1236(02)00065-4. |
[20] |
D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55.
|
[21] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Applied Mathematical Series, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[22] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications,, Gordon and Breach Science Publishers, (1993).
|
[23] |
L. C. Young, An integration of Höder type, connected with Stieltjes integration,, Acta Math., 67 (1936), 251.
doi: 10.1007/BF02401743. |
[24] |
M. Zähle, Integration with respect to fractal functions and stochastic calculus. I,, Probab. Theory Related Fields, 111 (1998), 333.
doi: 10.1007/s004400050171. |
show all references
References:
[1] |
M. Caruana and P. Friz, Partial differential equations driven by rough paths,, J. Differential Equations, 247 (2009), 140.
doi: 10.1016/j.jde.2009.01.026. |
[2] |
M. Caruana, P. Friz and H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 27.
doi: 10.1016/j.anihpc.2010.11.002. |
[3] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems,, Discrete and Continuous Dynamical Systems, 34 (2014), 79.
doi: 10.3934/dcds.2014.34.79. |
[4] |
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.
doi: 10.1214/10-AIHP392. |
[5] |
A. Deya, M. Gubinelli and S. Tindel, Non-linear rough heat equations,, Probab. Theory Relat. Fields, 153 (2012), 97.
doi: 10.1007/s00440-011-0341-z. |
[6] |
P. Friz and H. Oberhauser, On the splitting-up method for rough (partial) differential equations,, J. Differential Equations, 251 (2011), 316.
doi: 10.1016/j.jde.2011.02.009. |
[7] |
P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications,, Cambridge Studies of Advanced Mathematics, (2010).
doi: 10.1017/CBO9780511845079. |
[8] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion,, Discrete and Continuous Dynamical Systems, 14 (2010), 473.
doi: 10.3934/dcdsb.2010.14.473. |
[9] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Compensated fractional derivatives and stochastic evolution equations,, Comptes Rendus Mathématique, 350 (2012), 1037.
doi: 10.1016/j.crma.2012.11.007. |
[10] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameter $H\in (1/3,1/2]$,, , (). Google Scholar |
[11] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Lévy areas of Ornstein-Uhlenbeck processes in Hilbert spaces,, Studies in Systems, 30 (2015), 167.
doi: 10.1007/978-3-319-19075-4_10. |
[12] |
M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion,, International Journal of Bifurcation and Chaos, 20 (2010), 2761.
doi: 10.1142/S0218127410027349. |
[13] |
M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs,, Potential Anal., 25 (2006), 307.
doi: 10.1007/s11118-006-9013-5. |
[14] |
M. Gubinelli and S. Tindel, Rough Evolution Equations,, The Annals of Probability, 38 (2010), 1.
doi: 10.1214/08-AOP437. |
[15] |
M. Hinz and M. Zähle, Gradient type noises II-Systems of stochastic partial differential equations,, Journal of Functional Analysis, 256 (2009), 3192.
doi: 10.1016/j.jfa.2009.02.006. |
[16] |
Y. Hu and D. Nualart, Rough path analysis via fractional calculus,, Trans. Amer. Math. Soc., 361 (2009), 2689.
doi: 10.1090/S0002-9947-08-04631-X. |
[17] |
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary theory,, Graduate Studies in Mathematics, (1997).
|
[18] |
T. Lyons and Z. Qian, System control and rough paths,, Oxford Mathematical Monographs, (2002).
doi: 10.1093/acprof:oso/9780198506485.001.0001. |
[19] |
B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion,, J. Funct. Anal., 202 (2003), 277.
doi: 10.1016/S0022-1236(02)00065-4. |
[20] |
D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55.
|
[21] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Applied Mathematical Series, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[22] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications,, Gordon and Breach Science Publishers, (1993).
|
[23] |
L. C. Young, An integration of Höder type, connected with Stieltjes integration,, Acta Math., 67 (1936), 251.
doi: 10.1007/BF02401743. |
[24] |
M. Zähle, Integration with respect to fractal functions and stochastic calculus. I,, Probab. Theory Related Fields, 111 (1998), 333.
doi: 10.1007/s004400050171. |
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