Compensated fractional derivatives and stochastic evolution equations

María J. Garrido-Atienza; Kening Lu; Björn Schmalfuß

doi:10.1016/j.crma.2012.11.007

Partial Differential Equations/Probability Theory

Compensated fractional derivatives and stochastic evolution equations
[Dérivées fractionnaires compensées et équations dʼévolution stochastiques]

Présenté par : the Editorial Board

María J. Garrido-Atienza ¹ ; Kening Lu ² ; Björn Schmalfuß ³

¹ Dpto. EDAN, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain
² 346 TMCB, Brigham Young University, Provo, UT 84602, USA
³ Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 77043 Jena, Germany

Comptes Rendus. Mathématique, Volume 350 (2012) no. 23-24, pp. 1037-1042.

Résumé
Abstract

Dans cette Note, nous sommes intéressés à développer une théorie trajectorielle pour les solutions ‘mild’ dʼéquations dʼévolution stochastiques lorsque le bruit est β-Hölder continue pour $β \in (1 / 3, 1 / 2)$ . Selon la théorie ‘Rough Path’, les intégrales stochastiques liés à la solution des équations différentielles ordinaires contiennent des éléments dʼun espace de tenseurs. Grâce aux dérivées fractionnaires (compensées), on peut formuler une deuxième équation pour ce tenseur, pour lequel nous construisons un autre tenseur en fonction non seulement sur le bruit, mais aussi sur le semi-groupe. Nous formulons des conditions suffisantes pour lʼexistence et lʼunicité dʼune solution trajectorielle en utilisant le théoréme du point fixe de Banach lorsque des coefficients du système sont assez régulières.

We are interested in developing a pathwise theory for mild solutions of stochastic evolution equations when the noise path is β-Hölder continuous for $β \in (1 / 3, 1 / 2)$ . From the point of view of the Rough Path Theory, stochastic integrals related to the solution of ordinary differential equations contain area-elements from a tensor space. Based on (compensated) fractional derivatives we are able to derive a second mild equation for these area components. We formulate sufficient conditions for the existence and uniqueness of a pathwise mild solution by using the Banach fixed point theorem provided that the coefficients of the system are sufficiently regular.

Reçu le : 2012-08-06
Accepté le : 2012-11-12
Publié le : 2012-11-23

DOI : 10.1016/j.crma.2012.11.007

Affiliations des auteurs :

María J. Garrido-Atienza ¹ ; Kening Lu ² ; Björn Schmalfuß ³

¹ Dpto. EDAN, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain
² 346 TMCB, Brigham Young University, Provo, UT 84602, USA
³ Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 77043 Jena, Germany

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     author = {Mar{\'\i}a J. Garrido-Atienza and Kening Lu and Bj\"orn Schmalfu{\ss}},
     title = {Compensated fractional derivatives and stochastic evolution equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1037--1042},
     publisher = {Elsevier},
     volume = {350},
     number = {23-24},
     year = {2012},
     doi = {10.1016/j.crma.2012.11.007},
     language = {en},
}

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María J. Garrido-Atienza; Kening Lu; Björn Schmalfuß. Compensated fractional derivatives and stochastic evolution equations. Comptes Rendus. Mathématique, Volume 350 (2012) no. 23-24, pp. 1037-1042. doi : 10.1016/j.crma.2012.11.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.11.007/

Bibliographie
Cité par

[1] M. Caruana; P. Friz Partial differential equations driven by rough paths, J. Differential Equations, Volume 247 (2009) no. 1, pp. 140-173

[2] A. Deya; M. Gubinelli; S. Tindel Non-linear rough heat equations, Probab. Theory Related Fields, Volume 153 (2012) no. 1–2, pp. 97-147

[3] M.J. Garrido-Atienza; K. Lu; B. Schmalfuss Pathwise solutions to stochastic partial differential equations driven by fractional Brownian motions with Hurst parameters in $(1 / 3, 1 / 2]$ | arXiv

[4] M. Gubinelli; S. Tindel Rough evolution equations, Ann. Probab., Volume 38 (2010) no. 1, pp. 1-75

[5] Y. Hu; D. Nualart Rough path analysis via fractional calculus, Trans. Amer. Math. Soc., Volume 361 (2009) no. 5, pp. 2689-2718

[6] R.V. Kadison; J.R. Ringrose Fundamentals of the Theory of Operator Algebras: Elementary Theory, Graduate Studies in Mathematics, AMS, 1997

[7] A. Pazy Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Applied Mathematical Series, Springer-Verlag, Berlin, 1983

[8] J. Teichmann Another approach to some rough and stochastic partial differential equations, Stoch. Dyn., Volume 11 (2011) no. 2–3, pp. 535-550

[9] M. Zähle Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields, Volume 111 (1998) no. 3, pp. 333-374

Müfit Şan Complex variable approach to the analysis of a fractional differential equation in the real line, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 356 (2018) no. 3, pp. 293-300 | DOI:10.1016/j.crma.2018.01.008 | Zbl:1386.34014
María J. Garrido-Atienza; Kening Lu; Björn Schmalfuss Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H \in (1 / 3, 1 / 2]$ , SIAM Journal on Applied Dynamical Systems, Volume 15 (2016) no. 1, pp. 625-654 | DOI:10.1137/15m1030303 | Zbl:1336.60103
María 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 and Continuous Dynamical Systems. Series B, Volume 20 (2015) no. 8, pp. 2553-2581 | DOI:10.3934/dcdsb.2015.20.2553 | Zbl:1335.60111
Luu Hoang Duc; Björn Schmalfuß; Stefan Siegmund A note on the generation of random dynamical systems from fractional stochastic delay differential equations, Stochastics and Dynamics, Volume 15 (2015) no. 3, p. 13 (Id/No 1550018) | DOI:10.1142/s0219493715500185 | Zbl:1317.37099
Yong Chen; Hongjun Gao; María J. Garrido-Atienza; Björn Schmalfuss Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1 / 2$ and random dynamical systems, Discrete and Continuous Dynamical Systems, Volume 34 (2014) no. 1, pp. 79-98 | DOI:10.3934/dcds.2014.34.79 | Zbl:1321.37076
Hongjun Gao; María J. Garrido-Atienza; Björn Schmalfuss Random Attractors for Stochastic Evolution Equations Driven by Fractional Brownian Motion, SIAM Journal on Mathematical Analysis, Volume 46 (2014) no. 4, p. 2281 | DOI:10.1137/130930662
Jan Bártek; María J. Garrido-Atienza; Bohdan Maslowski Stochastic porous media equation driven by fractional Brownian motion, Stochastics and Dynamics, Volume 13 (2013) no. 4, p. 33 (Id/No 1350010) | DOI:10.1142/s021949371350010x | Zbl:1291.35453

Cité par 7 documents. Sources : Crossref, zbMATH

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