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Convergence of solutions of stochastic differential equations to the Arratia flow

Authors:

T. V. Malovichko

Affiliation:

(1) “Kyiv Polytechnic Institute” Ukrainian National Technical University, Kyiv, Ukraine

Abstract:

We consider the solution x _ε of the equation

$begin{array}{*{20}c} {{dx_{{text{ $ varepsilon $ }}} {left( {u,t} right)} = {intlimits_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. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1529–1538, November, 2008.

Keywords:

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