Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

$$\begin{array}{*{20}c} {{dx_{{\text{ $ \varepsilon $ }}} {\left( {u,t} \right)} = {\int\limits_\mathbb{R} {{\rm{\varphi}} _{{\text{ $ \varepsilon $ }}} {\left( {x_{{\text{ $ \varepsilon $ }}} {\left( {u,t} \right)} - r} \right)}W{\left( {dr,dt} \right)},} }}} \\ {{x_{{\text{ $ \varepsilon $ }}} {\left( {u,0} \right)} = u,}} \\ \end{array} $$

where W is a Wiener sheet on $\mathbb{R} \times {\left[ {0;1} \right]}$. In the case where φ_ε ² converges to pδ(⋅ −a ₁) + qδ(⋅ −a ₂), i.e., the limit function describing the influence of a random medium is singular at more than one point, we establish the weak convergence of (x _ε (u ₁,⋅), …, x _ε (u _d , ⋅)) as ε → 0₊ to (X(u ₁,⋅), …, X(u _d , ⋅)), where X is the Arratia flow.