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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  February 2019 Early access  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 and 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); $\alpha = 0.9$, $c = 1$
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