2017
Том 69
№ 5

All Issues

Convergence of solutions of stochastic differential equations to the Arratia flow

Malovichko T. V.

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Abstract

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).

English version (Springer): Ukrainian Mathematical Journal 60 (2008), no. 11, pp 1789-1802.

Citation Example: Malovichko T. V. Convergence of solutions of stochastic differential equations to the Arratia flow // Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1529–1538.

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