On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval

A. A. Dorogovtsev; M. B. Vovchanskii

doi:10.37863/umzh.v72i9.6279

A. A. Dorogovtsev Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv,and Nat. Techn. Univ. Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
M. B. Vovchanskii Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv https://orcid.org/0000-0002-6923-7503

DOI: https://doi.org/10.37863/umzh.v72i9.6279

Keywords: Brownian web, Arratia flow, Fractional Step Method, Splitting, Random Measure, Stochastic Flow, Stochastic Differential Equations

Abstract

UDC 519.21

We establish the rate of weak convergence in the fractional step method for the Arratia flow in terms of the Wasserstein distance between the images of the Lebesque measure under the action of the flow. We introduce finite-dimensional densities that describe sequences of collisions in the Arratia flow and derive an explicit expression for them. With the initial interval discretized, we also discuss the convergence of the corresponding approximations of the point measure associated with the Arratia flow in terms of such densities.

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