November  2015, 35(11): 5255-5272. doi: 10.3934/dcds.2015.35.5255

Stochastic Korteweg-de Vries equation driven by fractional Brownian motion

1. 

Department of Mathematics, Tongji University, Shanghai 200092, China

2. 

Institute of Applied Physics & Computational Math., Beijing 100088

Received  September 2013 Revised  May 2014 Published  May 2015

We consider the Cauchy problem for the Korteweg-de Vries equation driven by a cylindrical fractional Brownian motion (fBm) in this paper. With Hurst parameter $H\geq\frac{7}{16}$ of the fBm, we obtain the local existence results with initial value in classical Sobolev spaces $H^s$ with $s\geq -\frac{9}{16}$. Furthermore, we give the relation between the Hurst parameter $H$ and the index $s$ to the Sobolev spaces $H^s$, which finds out the regularity between the driven term fBm and the initial value for the stochastic Korteweg-de Vries equation.
Citation: Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
References:
[1]

E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes,, Annals of Probab., 29 (2001), 766.  doi: 10.1214/aop/1008956692.  Google Scholar

[2]

E. Alòs and D. Nualart, Stochastic calculus with respect to fractional Brownian motion,, Stoch. Stoch. Rep., 75 (2003), 129.  doi: 10.1080/1045112031000078917.  Google Scholar

[3]

F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications,, Probability and its Applications (New York), (2008).  doi: 10.1007/978-1-84628-797-8.  Google Scholar

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J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: The KdV equation,, Geom. Funct. Anal., 2 (1993), 107.   Google Scholar

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P. Caithamer, The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise,, Stoch. Dyn., 5 (2005), 45.  doi: 10.1142/S0219493705001286.  Google Scholar

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H. Y. Chang, C. Lien, S. Sukarto, S. Raychaudhury, J. Hill, E. K. Tsikis and K. E. Lonngren, Propagation of ion-acoustic solitons in a non-quiescent plasma,, Plasma Phys. Control. Fusion, 28 (1986), 675.  doi: 10.1088/0741-3335/28/4/005.  Google Scholar

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G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar

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A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Korteweg-de Veris equation,, J. Funct. Anal., 169 (1999), 532.  doi: 10.1006/jfan.1999.3484.  Google Scholar

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A. de Bouard, A. Debussche and Y. Tsutsumi, Periocic solutions of the Korteweg-de Veris equation driven by white noise,, SIAM J. Math. Anal., 36 (2004), 815.  doi: 10.1137/S0036141003425301.  Google Scholar

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T. E. Duncan, J. Jakubowski and B. Pasik-Duncan, Stochastic integration for fractional Brownian motion in a Hilbert space,, Stoch. Dyn., 6 (2006), 53.  doi: 10.1142/S0219493706001645.  Google Scholar

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T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Fractional Brownian motion and stochastic equations in Hilbert spaces,, Stoch. Dyn., 2 (2002), 225.  doi: 10.1142/S0219493702000340.  Google Scholar

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M. Erraoui, D. Nualart and Y. Ouknine, Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet,, Stoch. Dyn., 3 (2003), 121.  doi: 10.1142/S0219493703000681.  Google Scholar

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W. Grecksch and V. V. Ahn, A parabolic stochastic differential equation with fractional Brownian motion input,, Stat. Probab. Lett., 41 (1999), 337.  doi: 10.1016/S0167-7152(98)00147-3.  Google Scholar

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B. Guo and Z. Huo, The well-posedness of the Korteweg-de Vries-Benjamin-Ono equation,, J. Math. Anal. Appl., 295 (2004), 444.  doi: 10.1016/j.jmaa.2004.02.043.  Google Scholar

[15]

R. Herman, The stochastic, damped Korteweg-de Vries equation,, J. Phys. A., 23 (1990), 1063.  doi: 10.1088/0305-4470/23/7/014.  Google Scholar

[16]

Y. Hu, Heat equation with fractional white noise potential,, Appl. Math. Optim., 43 (2001), 221.  doi: 10.1007/s00245-001-0001-2.  Google Scholar

[17]

Y. Hu and D. Nualart, Stochastic heat equation driven by fractional noise and local time,, Probab. Theory Related Fields, 143 (2009), 285.  doi: 10.1007/s00440-007-0127-5.  Google Scholar

[18]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the Kdv equation,, J. Amer. Math. Soc., 9 (1996), 573.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

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C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,, Duke Math. J., 71 (1993), 1.  doi: 10.1215/S0012-7094-93-07101-3.  Google Scholar

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A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum,, C. R. (Doklady) Acad. URSS (N.S.), 26 (1940), 115.   Google Scholar

[21]

B. B. Mandelbrot, The Fractal Geometry of Nature,, W. H. Freeman and Co., (1982).   Google Scholar

[22]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications,, SIAM Rev., 10 (1968), 422.  doi: 10.1137/1010093.  Google Scholar

[23]

B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion,, J. Funct. Anl., 202 (2003), 277.  doi: 10.1016/S0022-1236(02)00065-4.  Google Scholar

[24]

Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes,, Lecture Notes in Mathematics, (1929).  doi: 10.1007/978-3-540-75873-0.  Google Scholar

[25]

D. Nualart, Malliavin Calculus and Related topics,, Probability and its Applications (New York), (1995).  doi: 10.1007/978-1-4757-2437-0.  Google Scholar

[26]

J. Printems, The stochastic Korteweg-de Vries equation in $L^2(\mathbb R)$,, J. Differ. Equations, 153 (1999), 338.  doi: 10.1006/jdeq.1998.3548.  Google Scholar

[27]

M. Scalerandi, A. Romano and C. A. Condat, Korteweg-de Vries solitons under additive stochastic perturbations,, Phys. Rev. E, 58 (1998), 4166.   Google Scholar

[28]

T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation,, Amer. J. Math., 123 (2001), 839.  doi: 10.1353/ajm.2001.0035.  Google Scholar

[29]

S. Tindel, C. A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion,, Probab. Theory Related Fields, 127 (2003), 186.  doi: 10.1007/s00440-003-0282-2.  Google Scholar

[30]

G. Wang, M. Zeng and B. Guo, Stochastic Burgers' equation driven by fractional Brownian motion,, J. Math. Anal. Appl., 371 (2010), 210.  doi: 10.1016/j.jmaa.2010.05.015.  Google Scholar

show all references

References:
[1]

E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes,, Annals of Probab., 29 (2001), 766.  doi: 10.1214/aop/1008956692.  Google Scholar

[2]

E. Alòs and D. Nualart, Stochastic calculus with respect to fractional Brownian motion,, Stoch. Stoch. Rep., 75 (2003), 129.  doi: 10.1080/1045112031000078917.  Google Scholar

[3]

F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications,, Probability and its Applications (New York), (2008).  doi: 10.1007/978-1-84628-797-8.  Google Scholar

[4]

J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: The KdV equation,, Geom. Funct. Anal., 2 (1993), 107.   Google Scholar

[5]

P. Caithamer, The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise,, Stoch. Dyn., 5 (2005), 45.  doi: 10.1142/S0219493705001286.  Google Scholar

[6]

H. Y. Chang, C. Lien, S. Sukarto, S. Raychaudhury, J. Hill, E. K. Tsikis and K. E. Lonngren, Propagation of ion-acoustic solitons in a non-quiescent plasma,, Plasma Phys. Control. Fusion, 28 (1986), 675.  doi: 10.1088/0741-3335/28/4/005.  Google Scholar

[7]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar

[8]

A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Korteweg-de Veris equation,, J. Funct. Anal., 169 (1999), 532.  doi: 10.1006/jfan.1999.3484.  Google Scholar

[9]

A. de Bouard, A. Debussche and Y. Tsutsumi, Periocic solutions of the Korteweg-de Veris equation driven by white noise,, SIAM J. Math. Anal., 36 (2004), 815.  doi: 10.1137/S0036141003425301.  Google Scholar

[10]

T. E. Duncan, J. Jakubowski and B. Pasik-Duncan, Stochastic integration for fractional Brownian motion in a Hilbert space,, Stoch. Dyn., 6 (2006), 53.  doi: 10.1142/S0219493706001645.  Google Scholar

[11]

T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Fractional Brownian motion and stochastic equations in Hilbert spaces,, Stoch. Dyn., 2 (2002), 225.  doi: 10.1142/S0219493702000340.  Google Scholar

[12]

M. Erraoui, D. Nualart and Y. Ouknine, Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet,, Stoch. Dyn., 3 (2003), 121.  doi: 10.1142/S0219493703000681.  Google Scholar

[13]

W. Grecksch and V. V. Ahn, A parabolic stochastic differential equation with fractional Brownian motion input,, Stat. Probab. Lett., 41 (1999), 337.  doi: 10.1016/S0167-7152(98)00147-3.  Google Scholar

[14]

B. Guo and Z. Huo, The well-posedness of the Korteweg-de Vries-Benjamin-Ono equation,, J. Math. Anal. Appl., 295 (2004), 444.  doi: 10.1016/j.jmaa.2004.02.043.  Google Scholar

[15]

R. Herman, The stochastic, damped Korteweg-de Vries equation,, J. Phys. A., 23 (1990), 1063.  doi: 10.1088/0305-4470/23/7/014.  Google Scholar

[16]

Y. Hu, Heat equation with fractional white noise potential,, Appl. Math. Optim., 43 (2001), 221.  doi: 10.1007/s00245-001-0001-2.  Google Scholar

[17]

Y. Hu and D. Nualart, Stochastic heat equation driven by fractional noise and local time,, Probab. Theory Related Fields, 143 (2009), 285.  doi: 10.1007/s00440-007-0127-5.  Google Scholar

[18]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the Kdv equation,, J. Amer. Math. Soc., 9 (1996), 573.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[19]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,, Duke Math. J., 71 (1993), 1.  doi: 10.1215/S0012-7094-93-07101-3.  Google Scholar

[20]

A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum,, C. R. (Doklady) Acad. URSS (N.S.), 26 (1940), 115.   Google Scholar

[21]

B. B. Mandelbrot, The Fractal Geometry of Nature,, W. H. Freeman and Co., (1982).   Google Scholar

[22]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications,, SIAM Rev., 10 (1968), 422.  doi: 10.1137/1010093.  Google Scholar

[23]

B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion,, J. Funct. Anl., 202 (2003), 277.  doi: 10.1016/S0022-1236(02)00065-4.  Google Scholar

[24]

Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes,, Lecture Notes in Mathematics, (1929).  doi: 10.1007/978-3-540-75873-0.  Google Scholar

[25]

D. Nualart, Malliavin Calculus and Related topics,, Probability and its Applications (New York), (1995).  doi: 10.1007/978-1-4757-2437-0.  Google Scholar

[26]

J. Printems, The stochastic Korteweg-de Vries equation in $L^2(\mathbb R)$,, J. Differ. Equations, 153 (1999), 338.  doi: 10.1006/jdeq.1998.3548.  Google Scholar

[27]

M. Scalerandi, A. Romano and C. A. Condat, Korteweg-de Vries solitons under additive stochastic perturbations,, Phys. Rev. E, 58 (1998), 4166.   Google Scholar

[28]

T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation,, Amer. J. Math., 123 (2001), 839.  doi: 10.1353/ajm.2001.0035.  Google Scholar

[29]

S. Tindel, C. A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion,, Probab. Theory Related Fields, 127 (2003), 186.  doi: 10.1007/s00440-003-0282-2.  Google Scholar

[30]

G. Wang, M. Zeng and B. Guo, Stochastic Burgers' equation driven by fractional Brownian motion,, J. Math. Anal. Appl., 371 (2010), 210.  doi: 10.1016/j.jmaa.2010.05.015.  Google Scholar

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