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:

[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 $H\in (1/3,1/2]$,, , ().
[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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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 $H\in (1/3,1/2]$,, , ().
[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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