Advection-diffusion equation on a half-line with boundary Lévy noise

Lena-Susanne Hartmann; Ilya Pavlyukevich

doi:10.3934/dcdsb.2018200

Article Contents

2019, Volume 24, Issue 2: 637-655. Doi: 10.3934/dcdsb.2018200

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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: April 30, 2017

Revised: January 31, 2018

Early access: June 2018

Published: January 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 60H15; Secondary: 35K10, 60F17, 60G51, 86A05.

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Figure 1. 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)

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Figure 2. 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 $\nu = 1$, $x = 1$

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Figure 3. 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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