Convergence of solutions of stochastic differential equations to the Arratia flow

  • T. V. Malovichko


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).
How to Cite
Malovichko, T. V. “Convergence of Solutions of Stochastic Differential Equations to the Arratia Flow”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, no. 11, Nov. 2008, pp. 1529–1538,
Research articles