# American Institute of Mathematical Sciences

• Previous Article
Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion
• DCDS-B Home
• This Issue
• Next Article
Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps
February 2019, 24(2): 637-655. doi: 10.3934/dcdsb.2018200

## 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

Received  May 2017 Revised  February 2018 Published  June 2018

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), [34]). We determine the closed form formulae for mild solutions of this equation with Dirichlet and Neumann noise and study approximations of these solutions by classical solutions obtained with the help of Wong-Zakai approximations of the driving Lévy process.

Citation: Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200
##### 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.
A sample path of an $\alpha$-stable Lévy subordinator $Z$ with ${\bf E} \text{e}^{-\lambda Z_1} = \text{e}^{-\lambda^\alpha}$ for $\alpha = 0.9$ (a); solutions $t\mapsto u_D(t, x)$ of equation (2.2) with Dirichlet boundary noise for $\nu = -1$, $x = 1$ (b) and $\nu = 1$, $x = 1$ (d); the concentration curve $x\mapsto u_D(t, x)$ for $\nu = 1$, $t = 55$ (c)
A sample path of a symmetric $\alpha$-stable Lévy process $Z$ with ${\bf{E}} \text{e}^{-\text{i} \lambda Z_1} = \text{e}^{-|\lambda|^\alpha}$ for $\alpha = 1.75$ (a); the solution $t\mapsto u_D(t, x)$ of equation (2.2) with Dirichlet boundary noise for $\nu = 1$, $x = 1$
The scales $c(x)$ of the limiting distribution in the Dirichlet case for $\nu = \pm1, 0$ (left), and the Neumann case for $\nu = -1$ (right); $\alpha = 0.9$, $c = 1$
 [1] Yeping Li, Jie Liao. Stability and $L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062 [2] Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068 [3] Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control & Related Fields, 2018, 8 (0) : 1-38. doi: 10.3934/mcrf.2019001 [4] Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009 [5] Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008 [6] Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao, Songsak Sriboonchitta. Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. Advances in Mathematics of Communications, 2018, 12 (2) : 231-262. doi: 10.3934/amc.2018016 [7] Wenqiang Zhao. Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2499-2526. doi: 10.3934/dcdsb.2018065 [8] Linglong Du. Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect. Networks & Heterogeneous Media, 2018, 13 (4) : 549-565. doi: 10.3934/nhm.2018025 [9] Hideaki Takagi. Times until service completion and abandonment in an M/M/$m$ preemptive-resume LCFS queue with impatient customers. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1701-1726. doi: 10.3934/jimo.2018028 [10] Chengxiang Wang, Li Zeng, Wei Yu, Liwei Xu. Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction. Inverse Problems & Imaging, 2018, 12 (3) : 545-572. doi: 10.3934/ipi.2018024 [11] Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109 [12] Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124 [13] Woocheol Choi, Yong-Cheol Kim. $L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5811-5834. doi: 10.3934/dcds.2018253 [14] Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159 [15] Qianying Xiao, Zuohuan Zheng. $C^1$ weak Palis conjecture for nonsingular flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1809-1832. doi: 10.3934/dcds.2018074 [16] Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252 [17] VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $p(x)$-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 [18] Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012 [19] Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058 [20] Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $p$-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070

2017 Impact Factor: 0.972

## Tools

Article outline

Figures and Tables