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February  2020, 40(2): 725-752. doi: 10.3934/dcds.2020059

Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system

Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, Plac Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland

* Corresponding author

Received  July 2018 Revised  August 2019 Published  November 2019

Fund Project: The first author is supported by NCN grant 2017/25/B/ST1/00051.

The dynamics is studied of an infinite collection of point particles placed in $ \mathbb{R}^d $, $ d\geq 1 $. The particles perform random jumps with mutual repulsion accompanied by random merging of pairs of particles. The states of the collection are probability measures on the corresponding configuration space. The main result is the proof of the existence of the Markov evolution of states for a bounded time horizon if the initial state is a sub-Poissonian measure. The proof is based on representing sub-Poissonian measures $ \mu $ by their correlation functions $ k_\mu $ and is done in two steps: (a) constructing an evolution $ k_{\mu_0} \to k_t $; (b) proving that $ k_t $ is the correlation function of a unique sub-Poissonian state $ \mu_t $.

Citation: Yuri Kozitsky, Krzysztof Pilorz. Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 725-752. doi: 10.3934/dcds.2020059
References:
[1]

R. A. Arratia, Coalescing Brownian Motion on the Line, , ProQuest LLC, Ann Arbor, MI, 1979.

[2]

J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05140-6.

[3]

J. Barańska and Yu. Kozitsky, The global evolution of states of a continuum Kawasaki model with repulsion, IMA J. Appl. Math., 83 (2018), 412-435.  doi: 10.1093/imamat/hxy006.

[4]

N. BerestyckiCh. Garban and A. Sen, Coalescing Brownian flows: A new approach, Ann. Probab., 43 (2015), 3177-3215.  doi: 10.1214/14-AOP957.

[5]

D. FinkelshteinYu. KondratievYu. Kozitsky and O. Kutoviy, The statistical dynamics of a spatial logistic model and the related kinetic equation, Math. Models Methods Appl. Sci., 25 (2015), 343-370.  doi: 10.1142/S0218202515500128.

[6]

N. L. Garcia and T. G. Kurtz, Spatial birth and death processes as solutions of stochastic equations, ALEA Lat. Am. J. Probab. Math. Stat., 1 (2006), 281-303. 

[7]

V. V. Konarovskii, On an infinite system of diffuising particles with coalescing, Theory Probab. Appl., 55 (2011), 134-144.  doi: 10.1137/S0040585X97984693.

[8]

V. V. Konarovskii and M. von Renesse, Modified massive Arratia flow and Wasserstein diffusion, Comm. Pure Appl. Math., 72 (2019), 764-800.  doi: 10.1002/cpa.21758.

[9]

Yu. Kondratiev and Yu. Kozitsky, The evolution of states in a spatial pupulation model, J. Dynam. Differential Equations, 30 (2018), 135-173.  doi: 10.1007/s10884-016-9526-6.

[10]

Yu. Kondratiev and T. Kuna, Harmonic analysis on configuration spaces. I. General theo, Inf. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2002), 201-233.  doi: 10.1142/S0219025702000833.

[11]

Yu. KondratievE. Lytvynov and M. Röckner, Non-equilibrium stochastic dynamics in continuum: The free case, Condens. Matt. Phys., 11 (2008), 701-721.  doi: 10.5488/CMP.11.4.701.

[12]

Y. Le Jan and O. Raimond, Flows, coalescernce and noise, Ann. Probab., 32 (2004), 1247-1315.  doi: 10.1214/009117904000000207.

[13]

S. Méléard and S. Roelly, Interacting branching measure processes, Stochastic Partial Differential Equations and Applications (Trento, 1990), 246–256, Pitman Res. Notes Math. Ser., 268, Longman Sci. Tech., Harlow, 1992.

[14]

I. Omelyan and Yu. Kozitsky, Spatially inhomogeneous population dynamics: Beyond the mean field approximation, J. Phys. A.: Math. Theor., 52 (2019), 305601 (18pp).

[15]

K. Pilorz, A kinetic equation for repulsive coalescing random jumps in continuum, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 70 (2016), 47–74. doi: 10.17951/a.2016.70.1.47.

[16]

E. Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Contiuum Mechanics, Springer, Berlin, 2009.

[17]

H. R. Thieme and J. Voigt, Stochastic semigroups: Their construction by perturbation and approximation, in: Positivity IV– Theory and Applications, Tech. Univ. Dresden, Dresden, (2006), 135–146.

show all references

References:
[1]

R. A. Arratia, Coalescing Brownian Motion on the Line, , ProQuest LLC, Ann Arbor, MI, 1979.

[2]

J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05140-6.

[3]

J. Barańska and Yu. Kozitsky, The global evolution of states of a continuum Kawasaki model with repulsion, IMA J. Appl. Math., 83 (2018), 412-435.  doi: 10.1093/imamat/hxy006.

[4]

N. BerestyckiCh. Garban and A. Sen, Coalescing Brownian flows: A new approach, Ann. Probab., 43 (2015), 3177-3215.  doi: 10.1214/14-AOP957.

[5]

D. FinkelshteinYu. KondratievYu. Kozitsky and O. Kutoviy, The statistical dynamics of a spatial logistic model and the related kinetic equation, Math. Models Methods Appl. Sci., 25 (2015), 343-370.  doi: 10.1142/S0218202515500128.

[6]

N. L. Garcia and T. G. Kurtz, Spatial birth and death processes as solutions of stochastic equations, ALEA Lat. Am. J. Probab. Math. Stat., 1 (2006), 281-303. 

[7]

V. V. Konarovskii, On an infinite system of diffuising particles with coalescing, Theory Probab. Appl., 55 (2011), 134-144.  doi: 10.1137/S0040585X97984693.

[8]

V. V. Konarovskii and M. von Renesse, Modified massive Arratia flow and Wasserstein diffusion, Comm. Pure Appl. Math., 72 (2019), 764-800.  doi: 10.1002/cpa.21758.

[9]

Yu. Kondratiev and Yu. Kozitsky, The evolution of states in a spatial pupulation model, J. Dynam. Differential Equations, 30 (2018), 135-173.  doi: 10.1007/s10884-016-9526-6.

[10]

Yu. Kondratiev and T. Kuna, Harmonic analysis on configuration spaces. I. General theo, Inf. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2002), 201-233.  doi: 10.1142/S0219025702000833.

[11]

Yu. KondratievE. Lytvynov and M. Röckner, Non-equilibrium stochastic dynamics in continuum: The free case, Condens. Matt. Phys., 11 (2008), 701-721.  doi: 10.5488/CMP.11.4.701.

[12]

Y. Le Jan and O. Raimond, Flows, coalescernce and noise, Ann. Probab., 32 (2004), 1247-1315.  doi: 10.1214/009117904000000207.

[13]

S. Méléard and S. Roelly, Interacting branching measure processes, Stochastic Partial Differential Equations and Applications (Trento, 1990), 246–256, Pitman Res. Notes Math. Ser., 268, Longman Sci. Tech., Harlow, 1992.

[14]

I. Omelyan and Yu. Kozitsky, Spatially inhomogeneous population dynamics: Beyond the mean field approximation, J. Phys. A.: Math. Theor., 52 (2019), 305601 (18pp).

[15]

K. Pilorz, A kinetic equation for repulsive coalescing random jumps in continuum, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 70 (2016), 47–74. doi: 10.17951/a.2016.70.1.47.

[16]

E. Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Contiuum Mechanics, Springer, Berlin, 2009.

[17]

H. R. Thieme and J. Voigt, Stochastic semigroups: Their construction by perturbation and approximation, in: Positivity IV– Theory and Applications, Tech. Univ. Dresden, Dresden, (2006), 135–146.

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