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Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion
Advection-diffusion equation on a half-line with boundary Lévy noise
Friedrich Schiller University Jena, School of Mathematics and Computer Science, Institute for Mathematics, Ernst-Abbe-Platz 2, 07743 Jena, Germany |
In this paper we study a one-dimensional linear advection-diffusion equation on a half-line driven by a Lévy boundary noise. The problem is motivated by the study of contaminant transport models under random sources (P. P. Wang and C. Zheng, Ground water, 43 (2005), [
References:
[1] |
E. Alòs and S. Bonaccorsi,
Stability for stochastic partial differential equations with Dirichlet white-noise boundary conditions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 5 (2002), 465-481.
doi: 10.1142/S0219025702000948. |
[2] |
E. Alòs and S. Bonaccorsi,
Stochastic partial differential equations with Dirichlet white-noise boundary conditions, Ann. Inst. H. Poincaré Probab. Statist, 38 (2002), 125-154.
doi: 10.1016/S0246-0203(01)01097-4. |
[3] |
A. V. Balakrishnan,
Applied Functional Analysis, vol. 3 of Applications of Mathematics, 2nd edition, Springer, New York, 1981. |
[4] |
H. Brezis,
Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[5] |
P. Brune, J. Duan and B. Schmalfuss,
Random dynamics of the Boussinesq system with dynamical boundary conditions, Stochastic Analysis and Applications, 27 (2009), 1096-1116.
doi: 10.1080/07362990902976546. |
[6] |
Z. Brzeźniak and F. Flandoli,
Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Processes and Their Applications, 55 (1995), 329-358.
doi: 10.1016/0304-4149(94)00037-T. |
[7] |
Z. Brzeźniak, B. Goldys, S. Peszat and F. Russo,
Second order PDEs with Dirichlet white noise boundary conditions, Journal of Evolution Equations, 15 (2015), 1-26.
doi: 10.1007/s00028-014-0246-2. |
[8] |
Z. Brzeźniak and S. Peszat, Hyperbolic equations with random boundary conditions, in Recent Development in Stochastic Dynamics and Stochastic Analysis (eds. J. Duan, S. Luo and C. Wang), vol. 8 of Interdisciplinary Mathematical Sciences, World Scientific, Singapore, 2010, 1-21.
doi: 10.1142/9789814277266_0001. |
[9] |
H. S. Carslaw and J. C. Jaeger,
Conduction of Heat in Solid, The Clarendon Press, Oxford University Press, New York, 1988. |
[10] |
A. Chaudhuri and M. Sekhar,
Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition, Stochastic Environmental Research and Risk Assessment, 21 (2006), 159-173.
doi: 10.1007/s00477-006-0053-6. |
[11] |
A. Chojnowska-Michalik,
On processes of Ornstein-Uhlenbeck type in Hilbert space, Stochastics, 21 (1987), 251-286.
doi: 10.1080/17442508708833459. |
[12] |
I. Chueshov and B. Schmalfuss,
Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential and Integral Equations, 17 (2004), 751-780.
|
[13] |
G. Da Prato and J. Zabczyk,
Evolution equations with white-noise boundary conditions, Stochastics and Stochastics Reports, 42 (1993), 167-182.
doi: 10.1080/17442509308833817. |
[14] |
G. Fabbri and B. Goldys,
An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise, SIAM Journal on Control and Optimization, 48 (2009), 1473-1488.
doi: 10.1137/070711529. |
[15] |
D. D. Haroske and H. Triebel,
Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbooks in Mathematics, European Mathematical Society, Zürich, 2008. |
[16] |
E. Hausenblas and P. A. Razafimandimby,
Controllability and qualitative properties of the solutions to SPDEs driven by boundary Lévy noise, Stochastic Partial Differential Equations: Analysis and Computations, 3 (2015), 221-271.
doi: 10.1007/s40072-015-0047-9. |
[17] |
W. A. Jury and H. Flühler,
Transport of chemicals through soil: Mechanisms, models, and field applications, Advances in agronomy, 47 (1992), 141-201.
doi: 10.1016/S0065-2113(08)60490-3. |
[18] |
A. Kreft and A. Zuber,
On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions, Chemical Engineering Science, 33 (1978), 1471-1480.
doi: 10.1016/0009-2509(78)85196-3. |
[19] |
J. L. Lions and E. Magenes,
Non-Homogeneous Boundary Value Problems and Applications I, vol. 181 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1972. |
[20] |
C. Man and C. W. Tsai,
Stochastic partial differential equation-based model for suspended sediment transport in surface water flows, Journal of Engineering Mechanics, 133 (2007), 422-430.
doi: 10.1061/(ASCE)0733-9399(2007)133:4(422). |
[21] |
F. Masiero,
A stochastic optimal control problem for the heat equation on the halfline with Dirichlet boundary-noise and boundary-control, Applied Mathematics & Optimization, 62 (2010), 253-294.
doi: 10.1007/s00245-010-9103-z. |
[22] |
S. Micu and E. Zuazua,
On the lack of null-controllability of the heat equation on the haf-line, Transactions of the American Mathematical Society, 353 (2000), 1635-1659.
doi: 10.1090/S0002-9947-00-02665-9. |
[23] |
J. C. Parker and M. T. van Genuchten,
Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport, Water Resources Research, 20 (1984), 866-872.
doi: 10.1029/WR020i007p00866. |
[24] |
I. Pavlyukevich and M. Riedle,
Non-standard Skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces, Stochastic Analysis and Applications, 33 (2015), 271-305.
doi: 10.1080/07362994.2014.988358. |
[25] |
I. Pavlyukevich and I. M. Sokolov,
One-dimensional space-discrete transport subject to Lévy perturbations, The Journal of Statistical Physics, 133 (2008), 205-215.
doi: 10.1007/s10955-008-9607-y. |
[26] |
A. Pazy,
Semigroups of Linear Operators and Applications toPartial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[27] |
S. Peszat and J. Zabczyk,
Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, vol. 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9780511721373. |
[28] |
A. D. Polyanin,
Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, FL, 2002. |
[29] |
M. Riedle,
Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes, Potential Analysis, 42 (2015), 809-838.
doi: 10.1007/s11118-014-9458-x. |
[30] |
A. V. Skorokhod,
Limit theorems for stochastic processes, Theory of Probability and its Applications, 1 (1956), 289-319.
|
[31] |
G. Tessitore and J. Zabczyk,
Wong-Zakai approximations of stochastic evolution equations, Journal of Evolution Equations, 6 (2006), 621-655.
doi: 10.1007/s00028-006-0280-9. |
[32] |
H. Triebel,
Theory of Function Spaces II, Monographs in Mathematics, Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0346-0419-2. |
[33] |
K. Twardowska,
On the approximation theorem of the Wong-Zakai type for the functional stochastic differential equations, Probability and Mathematical Statistics, 12 (1991), 319-334.
|
[34] |
P. P. Wang and C. Zheng,
Contaminant transport models under random sources, Ground Water, 43 (2005), 423-433.
|
[35] |
W. Whitt,
Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer, 2002. |
[36] |
E. Wong and M. Zakai,
On the convergence of ordinary integrals to stochastic integrals, The Annals of Mathematical Statistics, 36 (1965), 1560-1564.
doi: 10.1214/aoms/1177699916. |
[37] |
E. Wong and M. Zakai,
On the relation between ordinary and stochastic differential equations, International Journal of Engineering Science, 3 (1965), 213-229.
doi: 10.1016/0020-7225(65)90045-5. |
show all references
References:
[1] |
E. Alòs and S. Bonaccorsi,
Stability for stochastic partial differential equations with Dirichlet white-noise boundary conditions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 5 (2002), 465-481.
doi: 10.1142/S0219025702000948. |
[2] |
E. Alòs and S. Bonaccorsi,
Stochastic partial differential equations with Dirichlet white-noise boundary conditions, Ann. Inst. H. Poincaré Probab. Statist, 38 (2002), 125-154.
doi: 10.1016/S0246-0203(01)01097-4. |
[3] |
A. V. Balakrishnan,
Applied Functional Analysis, vol. 3 of Applications of Mathematics, 2nd edition, Springer, New York, 1981. |
[4] |
H. Brezis,
Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[5] |
P. Brune, J. Duan and B. Schmalfuss,
Random dynamics of the Boussinesq system with dynamical boundary conditions, Stochastic Analysis and Applications, 27 (2009), 1096-1116.
doi: 10.1080/07362990902976546. |
[6] |
Z. Brzeźniak and F. Flandoli,
Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Processes and Their Applications, 55 (1995), 329-358.
doi: 10.1016/0304-4149(94)00037-T. |
[7] |
Z. Brzeźniak, B. Goldys, S. Peszat and F. Russo,
Second order PDEs with Dirichlet white noise boundary conditions, Journal of Evolution Equations, 15 (2015), 1-26.
doi: 10.1007/s00028-014-0246-2. |
[8] |
Z. Brzeźniak and S. Peszat, Hyperbolic equations with random boundary conditions, in Recent Development in Stochastic Dynamics and Stochastic Analysis (eds. J. Duan, S. Luo and C. Wang), vol. 8 of Interdisciplinary Mathematical Sciences, World Scientific, Singapore, 2010, 1-21.
doi: 10.1142/9789814277266_0001. |
[9] |
H. S. Carslaw and J. C. Jaeger,
Conduction of Heat in Solid, The Clarendon Press, Oxford University Press, New York, 1988. |
[10] |
A. Chaudhuri and M. Sekhar,
Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition, Stochastic Environmental Research and Risk Assessment, 21 (2006), 159-173.
doi: 10.1007/s00477-006-0053-6. |
[11] |
A. Chojnowska-Michalik,
On processes of Ornstein-Uhlenbeck type in Hilbert space, Stochastics, 21 (1987), 251-286.
doi: 10.1080/17442508708833459. |
[12] |
I. Chueshov and B. Schmalfuss,
Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential and Integral Equations, 17 (2004), 751-780.
|
[13] |
G. Da Prato and J. Zabczyk,
Evolution equations with white-noise boundary conditions, Stochastics and Stochastics Reports, 42 (1993), 167-182.
doi: 10.1080/17442509308833817. |
[14] |
G. Fabbri and B. Goldys,
An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise, SIAM Journal on Control and Optimization, 48 (2009), 1473-1488.
doi: 10.1137/070711529. |
[15] |
D. D. Haroske and H. Triebel,
Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbooks in Mathematics, European Mathematical Society, Zürich, 2008. |
[16] |
E. Hausenblas and P. A. Razafimandimby,
Controllability and qualitative properties of the solutions to SPDEs driven by boundary Lévy noise, Stochastic Partial Differential Equations: Analysis and Computations, 3 (2015), 221-271.
doi: 10.1007/s40072-015-0047-9. |
[17] |
W. A. Jury and H. Flühler,
Transport of chemicals through soil: Mechanisms, models, and field applications, Advances in agronomy, 47 (1992), 141-201.
doi: 10.1016/S0065-2113(08)60490-3. |
[18] |
A. Kreft and A. Zuber,
On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions, Chemical Engineering Science, 33 (1978), 1471-1480.
doi: 10.1016/0009-2509(78)85196-3. |
[19] |
J. L. Lions and E. Magenes,
Non-Homogeneous Boundary Value Problems and Applications I, vol. 181 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1972. |
[20] |
C. Man and C. W. Tsai,
Stochastic partial differential equation-based model for suspended sediment transport in surface water flows, Journal of Engineering Mechanics, 133 (2007), 422-430.
doi: 10.1061/(ASCE)0733-9399(2007)133:4(422). |
[21] |
F. Masiero,
A stochastic optimal control problem for the heat equation on the halfline with Dirichlet boundary-noise and boundary-control, Applied Mathematics & Optimization, 62 (2010), 253-294.
doi: 10.1007/s00245-010-9103-z. |
[22] |
S. Micu and E. Zuazua,
On the lack of null-controllability of the heat equation on the haf-line, Transactions of the American Mathematical Society, 353 (2000), 1635-1659.
doi: 10.1090/S0002-9947-00-02665-9. |
[23] |
J. C. Parker and M. T. van Genuchten,
Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport, Water Resources Research, 20 (1984), 866-872.
doi: 10.1029/WR020i007p00866. |
[24] |
I. Pavlyukevich and M. Riedle,
Non-standard Skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces, Stochastic Analysis and Applications, 33 (2015), 271-305.
doi: 10.1080/07362994.2014.988358. |
[25] |
I. Pavlyukevich and I. M. Sokolov,
One-dimensional space-discrete transport subject to Lévy perturbations, The Journal of Statistical Physics, 133 (2008), 205-215.
doi: 10.1007/s10955-008-9607-y. |
[26] |
A. Pazy,
Semigroups of Linear Operators and Applications toPartial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[27] |
S. Peszat and J. Zabczyk,
Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, vol. 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9780511721373. |
[28] |
A. D. Polyanin,
Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, FL, 2002. |
[29] |
M. Riedle,
Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes, Potential Analysis, 42 (2015), 809-838.
doi: 10.1007/s11118-014-9458-x. |
[30] |
A. V. Skorokhod,
Limit theorems for stochastic processes, Theory of Probability and its Applications, 1 (1956), 289-319.
|
[31] |
G. Tessitore and J. Zabczyk,
Wong-Zakai approximations of stochastic evolution equations, Journal of Evolution Equations, 6 (2006), 621-655.
doi: 10.1007/s00028-006-0280-9. |
[32] |
H. Triebel,
Theory of Function Spaces II, Monographs in Mathematics, Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0346-0419-2. |
[33] |
K. Twardowska,
On the approximation theorem of the Wong-Zakai type for the functional stochastic differential equations, Probability and Mathematical Statistics, 12 (1991), 319-334.
|
[34] |
P. P. Wang and C. Zheng,
Contaminant transport models under random sources, Ground Water, 43 (2005), 423-433.
|
[35] |
W. Whitt,
Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer, 2002. |
[36] |
E. Wong and M. Zakai,
On the convergence of ordinary integrals to stochastic integrals, The Annals of Mathematical Statistics, 36 (1965), 1560-1564.
doi: 10.1214/aoms/1177699916. |
[37] |
E. Wong and M. Zakai,
On the relation between ordinary and stochastic differential equations, International Journal of Engineering Science, 3 (1965), 213-229.
doi: 10.1016/0020-7225(65)90045-5. |



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