Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters H\in (1/3,1/2]

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October 2015, 20(8): 2553-2581. doi: 10.3934/dcdsb.2015.20.2553

Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$

María J. Garrido–Atienza ^1, , Kening Lu ^2, and Björn Schmalfuss ^3,

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

Received November 2014 Revised March 2015 Published August 2015

Abstract
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In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting of a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which in particular includes white noise.

Keywords: Stochastic PDEs, Hilbert-space valued fractional Brownian motion, pathwise solutions..

Mathematics Subject Classification: Primary: 60H15; Secondary: 60H05, 60G22, 26A33, 26A4.

Citation: María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553

References:

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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[13]	M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs,, Potential Anal., 25 (2006), 307. doi: 10.1007/s11118-006-9013-5. Google Scholar
[14]	M. Gubinelli and S. Tindel, Rough Evolution Equations,, The Annals of Probability, 38 (2010), 1. doi: 10.1214/08-AOP437. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary theory,, Graduate Studies in Mathematics, (1997). Google Scholar
[18]	T. Lyons and Z. Qian, System control and rough paths,, Oxford Mathematical Monographs, (2002). doi: 10.1093/acprof:oso/9780198506485.001.0001. Google Scholar
[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. Google Scholar
[20]	D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55. Google Scholar
[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. Google Scholar
[22]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications,, Gordon and Breach Science Publishers, (1993). Google Scholar
[23]	L. C. Young, An integration of Höder type, connected with Stieltjes integration,, Acta Math., 67 (1936), 251. doi: 10.1007/BF02401743. Google Scholar
[24]	M. Zähle, Integration with respect to fractal functions and stochastic calculus. I,, Probab. Theory Related Fields, 111 (1998), 333. doi: 10.1007/s004400050171. Google Scholar

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[13]	M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs,, Potential Anal., 25 (2006), 307. doi: 10.1007/s11118-006-9013-5. Google Scholar
[14]	M. Gubinelli and S. Tindel, Rough Evolution Equations,, The Annals of Probability, 38 (2010), 1. doi: 10.1214/08-AOP437. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary theory,, Graduate Studies in Mathematics, (1997). Google Scholar
[18]	T. Lyons and Z. Qian, System control and rough paths,, Oxford Mathematical Monographs, (2002). doi: 10.1093/acprof:oso/9780198506485.001.0001. Google Scholar
[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. Google Scholar
[20]	D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55. Google Scholar
[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. Google Scholar
[22]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications,, Gordon and Breach Science Publishers, (1993). Google Scholar
[23]	L. C. Young, An integration of Höder type, connected with Stieltjes integration,, Acta Math., 67 (1936), 251. doi: 10.1007/BF02401743. Google Scholar
[24]	M. Zähle, Integration with respect to fractal functions and stochastic calculus. I,, Probab. Theory Related Fields, 111 (1998), 333. doi: 10.1007/s004400050171. Google Scholar

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