Convergence of solutions of stochastic differential equations to the Arratia flow
We consider the solution $x_{\varepsilon}$ of the equation $$dx_{\varepsilon}(u,t) = \int\limits_\mathbb{R}\varphi_{\varepsilon}(x_{\varepsilon}(u,t) - r) W(dr,dt), $$ $$x_{\varepsilon}(u,0) = u,$$ where $W$ is a Wiener sheet on $\mathbb{R} \times [0; 1].$ For the case where $\varphi_{\varepsilon}^...
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| Date: | 2008 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2008
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3266 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We consider the solution $x_{\varepsilon}$ of the equation
$$dx_{\varepsilon}(u,t) = \int\limits_\mathbb{R}\varphi_{\varepsilon}(x_{\varepsilon}(u,t) - r) W(dr,dt), $$
$$x_{\varepsilon}(u,0) = u,$$
where $W$ is a Wiener sheet on $\mathbb{R} \times [0; 1].$
For the case where $\varphi_{\varepsilon}^2$ converges to
$p \delta(\cdot - a_1) + q \delta(\cdot - a_2),$ i.e., where a boundary function describing the influence
of a random medium is singular more than at one point, we prove that the weak convergence of
$\left(x_{\varepsilon}(u_1, \cdot),...,x_{\varepsilon}(u_d, \cdot) \right)$ to
$\left(X(u_1, \cdot),...,X(u_d, \cdot) \right)$ takes place as $\varepsilon\rightarrow0_+$ (here, $X$ is the Arratia flow). |
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