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

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

Анотація

УДК 519.21

Про апроксимацiю асоцiйованих iз броунiвською сiткою точкових мiр за допомогою методу дробових крокiв та дискретизацiї початкового iнтервалу

Встановлено швидкiсть слабкої збiжностi в методi дробових крокiв для потоку Арратья в термiнах вiдстанi Вассерштейна мiж образами мiри Лебега пiд дiєю потоку. Введено скiнченновимiрнi щiльностi, що описують послiдовностi зiткнень в потоцi Арратья, та отримано явний вираз для них. Дослiджено збiжнiсть вiдповiдних апроксимацiй точкових мiр для потоку Арратья при дискретизацiї початкового iнтервалу.

Посилання

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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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