# American Institute of Mathematical Sciences

• Previous Article
Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps
• DCDS-B Home
• This Issue
• Next Article
Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion
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:

show all references

##### References:
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] Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020207 [2] Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $L^1$-control in coefficients for quasi-linear Dirichlet boundary value problems with $BMO$-anisotropic $p$-Laplacian. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020021 [3] El Haj Laamri, Michel Pierre. Stationary reaction-diffusion systems in $L^1$ revisited. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020355 [4] Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $(L^2,L^\gamma\cap H_0^1)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension. Electronic Research Archive, 2020, 28 (3) : 1357-1374. doi: 10.3934/era.2020072 [5] Anhui Gu. Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5737-5767. doi: 10.3934/dcdsb.2019104 [6] 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 [7] Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4513-4545. doi: 10.3934/dcdsb.2019154 [8] Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $1$-$d$ coupled wave equations. Mathematical Control & Related Fields, 2019  doi: 10.3934/mcrf.2020015 [9] Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $\bf{M/G/1}$ fault-tolerant machining system with imperfection. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019096 [10] 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 [11] Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $L_1$ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037 [12] Kaikai Cao, Youming Liu. Uncompactly supported density estimation with $L^{1}$ risk. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4007-4022. doi: 10.3934/cpaa.2020177 [13] Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $l_1$ norm measure. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2425-2437. doi: 10.3934/jimo.2019061 [14] Lingyan Cheng, Ruinan Li, Liming Wu. Exponential convergence in the Wasserstein metric $W_1$ for one dimensional diffusions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5131-5148. doi: 10.3934/dcds.2020222 [15] Tuan Anh Dao, Michael Reissig. $L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222 [16] Zalman Balanov, Yakov Krasnov. On good deformations of $A_m$-singularities. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1851-1866. doi: 10.3934/dcdss.2019122 [17] Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $m$-ary $n$-cubes $Q^m_n$. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020079 [18] Dandan Ma, Ji Shu, Ling Qin. Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020100 [19] 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 [20] Jiao Song, Jiang-Lun Wu, Fangzhou Huang. First jump time in simulation of sampling trajectories of affine jump-diffusions driven by $\alpha$-stable white noise. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4127-4142. doi: 10.3934/cpaa.2020184

2019 Impact Factor: 1.27