TY - JOUR AU - T. V. Malovichko PY - 2008/11/25 Y2 - 2024/03/29 TI - Convergence of solutions of stochastic differential equations to the Arratia flow JF - Ukrains’kyi Matematychnyi Zhurnal JA - Ukr. Mat. Zhurn. VL - 60 IS - 11 SE - Research articles DO - UR - https://umj.imath.kiev.ua/index.php/umj/article/view/3266 AB - 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). ER -