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Optimal surviving strategy for drifted Brownian motions with absorption. (English) Zbl 1429.60075
Summary: We study the “up the river” problem formulated by D. Aldous [“Up to the river”, Preprint], where a unit drift is distributed among a finite collection of Brownian particles on \(\mathbb{R}_{+}\), which are annihilated once they reach the origin. Starting \(K\) particles at \(x=1\), we prove Aldous’ conjecture [loc. cit.] that the “push-the-laggard” strategy of distributing the drift asymptotically (as \(K\rightarrow\infty\)) maximizes the total number of surviving particles, with approximately \(\frac{4}{\sqrt{\pi}}\sqrt{K}\) surviving particles. We further establish the hydrodynamic limit of the particle density, in terms of a two-phase partial differential equation (PDE) with a moving boundary, by utilizing certain integral identities and coupling techniques.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
35Q70 PDEs in connection with mechanics of particles and systems of particles
82C22 Interacting particle systems in time-dependent statistical mechanics
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[1] Aldous, D. (2002). Unpublished. Available at http://www.stat.berkeley.edu/ aldous/Research/OP/river.pdf.
[2] Banner, A. D., Fernholz, R. and Karatzas, I. (2005). Atlas models of equity markets. Ann. Appl. Probab.15 2296-2330. · Zbl 1099.91056
[3] Cabezas, M., Dembo, A., Sarantsev, A. and Sidoravicius, V. Brownian particles of rank-dependent drifts: Out of equilibrium behavior. Preprint. Available at arXiv:1708.01918. · Zbl 1415.60113
[4] Chatterjee, S. and Pal, S. (2010). A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Related Fields 147 123-159. · Zbl 1188.60049
[5] Chatterjee, S. and Pal, S. (2011). A combinatorial analysis of interacting diffusions. J. Theoret. Probab.24 939-968. · Zbl 1236.60093
[6] Dembo, A., Shkolnikov, M., Varadhan, S. R. S. and Zeitouni, O. (2016). Large deviations for diffusions interacting through their ranks. Comm. Pure Appl. Math.69 1259-1313. · Zbl 1341.60010
[7] Dembo, A. and Tsai, L.-C. (2017). Equilibrium fluctuation of the Atlas model. Ann. Probab.45 4529-4560. DOI:10.1214/16-AOP1171.
[8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York. · Zbl 0219.60003
[9] Fernholz, E. R. (2002). Stochastic Portfolio Theory. Stochastic Modelling and Applied Probability. Applications of Mathematics (New York) 48. Springer, New York. · Zbl 1049.91067
[10] Friedman, A. (1982). Variational Principles and Free-Boundary Problems. Wiley, New York. · Zbl 0564.49002
[11] Han, W. (2013). Available at http://hanweijian.com/research/2013-research-projects/random-particle-motion/.
[12] Hernández, F., Jara, M. and Valentim, F. J. (2017). Equilibrium fluctuations for a discrete Atlas model. Stochastic Process. Appl.127 783-802. · Zbl 1355.60121
[13] Ichiba, T. and Karatzas, I. (2010). On collisions of Brownian particles. Ann. Appl. Probab.20 951-977. · Zbl 1235.60111
[14] Ichiba, T., Karatzas, I. and Shkolnikov, M. (2013). Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156 229-248. · Zbl 1302.60092
[15] Ichiba, T., Papathanakos, V., Banner, A., Karatzas, I. and Fernholz, R. (2011). Hybrid atlas models. Ann. Appl. Probab.21 609-644. · Zbl 1230.60046
[16] Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge. Reprint of the 1990 original. · Zbl 0865.60043
[17] McKean, H. P. and Shepp, L. A. (2005). The advantage of capitalism vs. socialism depends on the criterion. J. Math. Sci.139 6589-6594. DOI:10.1007/s10958-006-0374-5. · Zbl 1121.91068
[18] Pal, S. and Pitman, J. (2008). One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab.18 2179-2207. · Zbl 1166.60061
[19] Pal, S. and Shkolnikov, M. (2014). Concentration of measure for Brownian particle systems interacting through their ranks. Ann. Appl. Probab.24 1482-1508. · Zbl 1297.82023
[20] Sarantsev, A. (2017). Infinite systems of competing Brownian particles. Ann. Inst. Henri Poincaré Probab. Stat.53 22792-2315. DOI:10.1214/16-AIHP791.
[21] Sarantsev, A. (2018). Comparison techniques for competing Brownian particles. J. Theoret. Probab. To appear. Available at arXiv:1305.1653. · Zbl 1429.60070
[22] Shkolnikov, M. (2011). Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab.21 1911-1932. · Zbl 1238.60113
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