Convergence of solutions of stochastic differential equations to the Arratia flow - Malovichko, T. V. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
We consider the solution $x_{ε}$ of the equation$$dx_{ε}(u,t) = \int\limits_\mathbb{R}φ_{ε}(x_{ε}(u,t) - r) W(dr,dt), $$$$x_{ε}(u,0) = u,$$where $W$ is a Wiener sheet on $\mathbb{R} × [0; 1].$ For the case where $φ_{ε}^2$ converges to$p δ(· - a_1) + q δ(· - a_2),$ 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$\left(x_{ε}(u_1, ·),...,x_{ε}(u_d, ·) \right)$ to$\left(X(u_1, ·),...,X(u_d, ·) \right)$ takes place as $ε→0_+$ (here, $X$ is the Arratia flow).