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) = _Rφ_ε(x_ε(u,t) - r) W(dr,dt), x_ε(u,0) = u, where W is a Wiener sheet on 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 (x_ε(u_1, ·),...,x_ε(u_d, ·) ) to (X(u_1, ·),...,X(u_d, ·) ) takes place as ε→0_+ (here, X is the Arratia flow).