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 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
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.

References

A. Bensoussan, R. Glowinski, A. Răşcanu, Approximation of some stochastic differential equations by the splitting up method, Appl. Math. and Optim., 25, no. 1, 81 – 106 (1992), https://doi.org/10.1007/BF01184157

A. A. Dorogovtsev, M. B. Vovchanskii, Arratia flow with drift and Trotter formula for Brownian web, Commun.Stoch. Anal., 12, No. 1, 89 – 105 (2018), https://doi.org/10.31390/cosa.12.1.07

A. A. Dorogovtsev, Measure-valued processes and stochastic flows, Proc. Inst. Math. NAS Ukraine, Math. and Appl., 66, Kiev (2007) 290 pp. ISBN: 978-966-02-4540-2 (in Russian)

A. A. Dorogovtsev, V. V. Fomichov, The rate of weak convergence of the $n$-point motions of Harris flows, Dynam. Syst. and Appl., 25, No. 3, 377 – 392 (2016)

A. A. Dorogovtsev, Ia. A. Korenovska, Some random integral operators related to a point processes, Theory Stoch. Process., 22(38), No. 1, 16 – 21 (2017)

A. A. Dorogovtsev, Ia. A. Korenovska, Essential sets for random operators constructed from an Arratia flow, Commun. Stoch. Anal., 11, No. 3, 301 – 312 (2017), https://doi.org/10.31390/cosa.11.3.03

V. Fomichov, The distribution of the number of clusters in the Arratia flow, Commun. Stoch. Anal., 10, No. 3, 257 – 270 (2016), https://doi.org/10.31390/cosa.10.3.01

L. R. Fontes, C. M. Newman, The full Brownian web as scaling limit of stochastic flows, Stoch. Dyn., 6, No. 2, 213 – 228 (2006), https://doi.org/10.1142/S0219493706001724

L. R. G. Fontes, M. Isopi, C. M. Newman, K. Ravishankar, The Brownian web: characterization and convergence, Ann. Probab., 32, No. 4, 2857 – 2883 (2004), https://doi.org/10.1214/009117904000000568

N. Yu. Goncharuk, P. Kotelenez, Fractional step method for stochastic evolution equations, Stoch. Process. and Appl., 73, No. 1, 1 – 45 (1998), https://doi.org/10.1016/S0304-4149(97)00079-3

E. V. Glinyanaya, Semigroups of $m$-point motions of the Arratia flow, and binary forests, Theory Stoch. Process., 19(35), No. 2, 31 – 41 (2014)

I. I. Gikhman, A. V. Skorokhod, Introduction to the theory of random processes, Dover Books Math., Dover Publ. (1996). xiv + 516 pp. ISBN: 0-486-69387-2

O. Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications, Springer-Verlag, New York (2002). xx+638 pp. ISBN: 0-387-95313-2, https://doi.org/10.1007/978-1-4757-4015-8

R. Munasinghe, R. Rajesh, R. Tribe, O. Zaboronski, Multi-scaling of the n-point density function for coalescing Brownian motions, Commun. Math. Phys., 268, no. 3, 717 – 725 (2006), https://doi.org/10.1007/s00220-006-0110-5

R. Tribe, O. Zaboronski, Pfaffian formulae for one dimensional coalescing and annihilating systems, Electron. J. Probab., 16, no. 76, 2080 – 2103 (2011), https://doi.org/10.1214/EJP.v16-942

C. Villani, Topics in optimal transportation, Amer. Math. Soc., Grad. Stud. Math., 58 (2003).xvi+370 pp. ISBN: 0-8218-3312-X, https://doi.org/10.1090/gsm/058

M. B. Vovchanskii, Convergence of solutions of SDEs to Harris flows, Theory Stoch. Process., 23(39), No. 2, 80 – 91 (2018), https://arxiv.org/abs/1909.06291

Published
22.09.2020
How to Cite
DorogovtsevA. A., and VovchanskiiM. B. “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”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 9, Sept. 2020, pp. 1179-94, doi:10.37863/umzh.v72i9.6279.
Section
Research articles