## Electronic Journal of Probability

### On asymptotic behavior of the modified Arratia flow

Vitalii Konarovskyi

#### Abstract

We study asymptotic properties of the system of interacting diffusion particles on the real line which transfer a mass [20]. The system is a natural generalization of the coalescing Brownian motions [3, 25]. The main difference is that diffusion particles coalesce summing their mass and changing their diffusion rate inversely proportional to the mass. First we construct the system in the case where the initial mass distribution has the moment of the order greater then two as an $L_2$-valued martingale with a suitable quadratic variation. Then we find the relationship between the asymptotic behavior of the particles and local properties of the mass distribution at the initial time.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 19, 31 pp.

Dates
Accepted: 30 January 2017
First available in Project Euclid: 18 February 2017

https://projecteuclid.org/euclid.ejp/1487386997

Digital Object Identifier
doi:10.1214/17-EJP34

Mathematical Reviews number (MathSciNet)
MR3622889

Zentralblatt MATH identifier
1358.82013

#### Citation

Konarovskyi, Vitalii. On asymptotic behavior of the modified Arratia flow. Electron. J. Probab. 22 (2017), paper no. 19, 31 pp. doi:10.1214/17-EJP34. https://projecteuclid.org/euclid.ejp/1487386997

#### References

• [1] Alberto Alonso and Fernando Brambila-Paz, $L^p$-continuity of conditional expectations, J. Math. Anal. Appl. 221 (1998), no. 1, 161–176.
• [2] Sebastian Andres and Max-K. von Renesse, Particle approximation of the Wasserstein diffusion, J. Funct. Anal. 258 (2010), no. 11, 3879–3905.
• [3] Richard Alejandro Arratia, Coalescing Brownian Motions on the Line, ProQuest LLC, Ann Arbor, MI, 1979, Thesis (Ph.D.)–The University of Wisconsin - Madison.
• [4] Patrick Billingsley, Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication.
• [5] P. P. Chernega, Local time at zero for Arratia flow, Ukrainian Math. J. 64 (2012), no. 4, 616–633.
• [6] Alexander S. Cherny and Hans-Jürgen Engelbert, Singular stochastic differential equations, Lecture Notes in Mathematics, vol. 1858, Springer-Verlag, Berlin, 2005.
• [7] Donald A. Dawson, Measure-valued Markov processes, Lecture Notes in Math., vol. 1541, Springer, Berlin, 1993.
• [8] A. A. Dorogovtsev, A. V. Gnedin, and M. B. Vovchanskii, Iterated logarithm law for sizes of clusters in Arratia flow, Theory Stoch. Process. 18 (2012), no. 2, 1–7.
• [9] A. A. Dorogovtsev and I. I. Nishchenko, An analysis of stochastic flows, Commun. Stoch. Anal. 8 (2014), no. 3, 331–342.
• [10] A. A. Dorogovtsev and O. V. Ostapenko, Large deviations for flows of interacting Brownian motions, Stoch. Dyn. 10 (2010), no. 3, 315–339.
• [11] Andrey A. Dorogovtsev, One Brownian stochastic flow, Theory Stoch. Process. 10 (2004), nos 3–4, 21–25.
• [12] Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986, Characterization and convergence.
• [13] Leszek Gawarecki and Vidyadhar Mandrekar, Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations, Probability and its Applications (New York), Springer, Heidelberg, 2011.
• [14] Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981.
• [15] Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 2003.
• [16] Adam Jakubowski, On the Skorokhod topology, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 3, 263–285.
• [17] Olav Kallenberg, Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002.
• [18] V. V. Konarovskii, The martingale problem for a measure-valued process with heavy diffusing particles, Theory Stoch. Process. 17 (2011), no. 1, 50–60.
• [19] V. V. Konarovskiĭ, On an infinite system of diffusing particles with coalescing, Teor. Veroyatn. Primen. 55 (2010), no. 1, 157–167.
• [20] V. Konarovskyi, A system of coalescing heavy diffusion particles on the real line, To appear in Ann. Probab. (2014), arXiv:1408.0628v3.
• [21] V. Konarovskyi and M.-K. von Renesse, Modified Arratia flow and Wasserstain diffusion, (2015), arXiv:1504.00559v3.
• [22] V. V. Konarovs’kyi, System of sticking diffusion particles of variable mass, Ukrainian Math. J. 62 (2010), no. 1, 97–113.
• [23] V. V. Konarovskyi, Large deviations principle for finite system of heavy diffusion particles, Theory Stoch. Process. 19 (2014), no. 1, 37–45.
• [24] Vitalii Konarovskyi, Mathematical model of heavy diffusion particles system with drift, Commun. Stoch. Anal. 7 (2013), no. 4, 591–605.
• [25] Yves Le Jan and Olivier Raimond, Flows, coalescence and noise, Ann. Probab. 32 (2004), no. 2, 1247–1315.
• [26] Robert S. Liptser and Albert N. Shiryaev, Statistics of random processes. I, expanded ed., Applications of Mathematics (New York), vol. 5, Springer-Verlag, Berlin, 2001, General theory, Translated from the 1974 Russian original by A. B. Aries, Stochastic Modelling and Applied Probability.
• [27] Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999.
• [28] Alexander Shamov, Short-time asymptotics of one-dimensional Harris flows, Commun. Stoch. Anal. 5 (2011), no. 3, 527–539.
• [29] Wilhelm Stannat, Two remarks on the Wasserstein Dirichlet form, Seminar on Stochastic Analysis, Random Fields and Applications VII, Progr. Probab., vol. 67, Birkhäuser/Springer, Basel, 2013, pp. 235–255.
• [30] Karl-Theodor Sturm, A monotone approximation to the Wasserstein diffusion, Singular phenomena and scaling in mathematical models, Springer, Cham, 2014, pp. 25–48.
• [31] Max-K. von Renesse and Karl-Theodor Sturm, Entropic measure and Wasserstein diffusion, Ann. Probab. 37 (2009), no. 3, 1114–1191.