The Annals of Applied Probability

Stochastic particle approximation of the Keller–Segel equation and two-dimensional generalization of Bessel processes

Nicolas Fournier and Benjamin Jourdain

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We are interested in the two-dimensional Keller–Segel partial differential equation. This equation is a model for chemotaxis (and for Newtonian gravitational interaction). When the total mass of the initial density is one, it is known to exhibit blow-up in finite time as soon as the sensitivity $\chi$ of bacteria to the chemo-attractant is larger than $8\pi$. We investigate its approximation by a system of $N$ two-dimensional Brownian particles interacting through a singular attractive kernel in the drift term.

In the very subcritical case $\chi<2\pi$, the diffusion strongly dominates this singular drift: we obtain existence for the particle system and prove that its flow of empirical measures converges, as $N\to\infty$ and up to extraction of a subsequence, to a weak solution of the Keller–Segel equation.

We also show that for any $N\ge2$ and any value of $\chi>0$, pairs of particles do collide with positive probability: the singularity of the drift is indeed visited. Nevertheless, when $\chi<2\pi N$, it is possible to control the drift and obtain existence of the particle system until the first time when at least three particles collide. We check that this time is a.s. infinite, so that global existence holds for the particle system, if and only if $\chi\leq8\pi(N-2)/(N-1)$.

Finally, we remark that in the system with $N=2$ particles, the difference between the two positions provides a natural two-dimensional generalization of Bessel processes, which we study in details.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 5 (2017), 2807-2861.

Dates
Received: July 2015
Revised: October 2016
First available in Project Euclid: 3 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1509696035

Digital Object Identifier
doi:10.1214/16-AAP1267

Mathematical Reviews number (MathSciNet)
MR3719947

Zentralblatt MATH identifier
06822206

Subjects
Primary: 65C35: Stochastic particle methods [See also 82C80] 35K55: Nonlinear parabolic equations 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Keller–Segel equation stochastic particle systems propagation of chaos Bessel processes

Citation

Fournier, Nicolas; Jourdain, Benjamin. Stochastic particle approximation of the Keller–Segel equation and two-dimensional generalization of Bessel processes. Ann. Appl. Probab. 27 (2017), no. 5, 2807--2861. doi:10.1214/16-AAP1267. https://projecteuclid.org/euclid.aoap/1509696035


Export citation

References

  • [1] Blanchet, A., Dolbeault, J. and Perthame, B. (2006). Two-dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions. Electron. J. Differential Equations 32.
  • [2] Bossy, M. and Talay, D. (1996). Convergence rate for the approximation of the limit law of weakly interacting particles: Application to the Burgers equation. Ann. Appl. Probab. 6 818–861.
  • [3] Carrillo, J. A., Lisini, S. and Mainini, E. (2014). Uniqueness for Keller–Segel-type chemotaxis models. Discrete Contin. Dyn. Syst. 34 1319–1338.
  • [4] Cattiaux, P. and Pédèches, L. (2016). The 2-D stochastic Keller–Segel particle model: Existence and uniqueness. ALEA Lat. Am. J. Probab. Math. Stat. 13 447–463.
  • [5] Cepa, E. and Lepingle, D. (2001). Brownian particles with electrostatic repulsion on the circle: Dyson’s model for unitary random matrices revisited. ESAIM Probab. Stat. 5 203–224.
  • [6] Dolbeault, J. and Schmeiser, C. (2009). The two-dimensional Keller–Segel model after blow-up. Discrete Contin. Dyn. Syst. 25 109–121.
  • [7] Egaña, G. and Mischler, S. (2016). Uniqueness and long time asymptotic for the Keller–Segel equation: The parabolic-elliptic case. Arch. Ration. Mech. Anal. 220 1159–1194.
  • [8] Fatkullin, I. (2013). A study of blow-ups in the Keller–Segel model of chemotaxis. Nonlinearity 26 81–94.
  • [9] Fournier, N. and Hauray, M. (2016). Propagation of chaos for the Landau equation with moderately soft potentials. Ann. Probab. 44 3581–3660.
  • [10] Fournier, N., Hauray, M. and Mischler, S. (2014). Propagation of chaos for the 2D viscous vortex model. J. Eur. Math. Soc. (JEMS) 16 1423–1466.
  • [11] Fukushima, M. (1980). Dirichlet Forms and Markov Processes. North-Holland, Amsterdam.
  • [12] Godinho, D. and Quininao, C. (2015). Propagation of chaos for a subcritical Keller–Segel model. Ann. Inst. Henri Poincaré Probab. Stat. 51 965–992.
  • [13] Haškovec, J. and Schmeiser, C. (2009). Stochastic particle approximation for measure valued solutions of the 2D Keller–Segel system. J. Stat. Phys. 135 133–151.
  • [14] Haškovec, J. and Schmeiser, C. (2011). Convergence of a stochastic particle approximation for measure solutions of the 2D Keller–Segel system. Comm. Partial Differential Equations 36 940–960.
  • [15] Hauray, M. and Jabin, P.-E. (2015). Particle approximation of Vlasov equations with singular forces: Propagation of chaos. Ann. Sci. Éc. Norm. Supér. (4) 48 891–940.
  • [16] Herrero, M. A. and Velázquez, J. J. L. (1996). Singularity patterns in a chemotaxis model. Math. Ann. 306 583–623.
  • [17] Horstmann, D. (2003). From 1970 until present: The Keller–Segel model in chemotaxis and its consequences I. Jahresber. Dtsch. Math.-Ver. 105 103–165.
  • [18] Horstmann, D. (2004). From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. II. Jahresber. Dtsch. Math.-Ver. 106 51–69.
  • [19] Jäger, W. and Luckhaus, S. (1992). On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329 819–824.
  • [20] Jourdain, B. (2000). Diffusion processes associated with nonlinear evolution equations for signed measures. Methodol. Comput. Appl. Probab. 2 69–91.
  • [21] Jourdain, B. and Reygner, J. (2016). A multitype sticky particle construction of Wasserstein stable semigroups solving one-dimensional diagonal hyperbolic systems with large monotonic data. J. Hyperbolic Differ. Equ. 13 441–602.
  • [22] Kac, M. (1956). Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 19541955, III. Univ. California Press, Berkeley, CA.
  • [23] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [24] Keller, E. F. and Segel, L. A. (1970). Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26 399–415.
  • [25] Khoshnevisan, D. (1994). Exact rates of convergence to Brownian local times. Ann. Probab. 22 1295–1330.
  • [26] Krylov, N. V. and Röckner, M. (2005). Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 154–196.
  • [27] Marchioro, C. and Pulvirenti, M. (1982). Hydrodynamics in two dimensions and vortex theory. Comm. Math. Phys. 84 483–503.
  • [28] McKean, H. P. Jr. (1967). Propagation of chaos for a class of non-linear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967) 41–57. Air Force Office Sci. Res., Arlington, VA.
  • [29] Méléard, S. (1996). Asymptotic behaviour of some interacting particle systems; McKean–Vlasov and Boltzmann models. In Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Math. 1627 42–95. Springer, Berlin.
  • [30] Mischler, S. and Mouhot, C. (2013). Kac’s program in kinetic theory. Invent. Math. 193 1–147.
  • [31] Osada, H. (1985). A stochastic differential equation arising from the vortex problem. Proc. Japan Acad. Ser. A Math. Sci. 61 333–336.
  • [32] Osada, H. (1986). Propagation of chaos for the two-dimensional Navier–Stokes equation. Proc. Japan Acad. Ser. A Math. Sci. 62 8–11.
  • [33] Osada, H. (1987). Propagation of chaos for the two-dimensional Navier–Stokes equation. In Probabilistic Methods in Mathematical Physics (Katata/Kyoto, 1985) 303–334. Academic Press, Boston, MA.
  • [34] Patlak, C. S. (1953). Random walk with persistence and external bias. Bull. Math. Biophys. 15 311–338.
  • [35] Perthame, B. (2004). PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic. Appl. Math. 49 539–564.
  • [36] Revuz, D. and Yor, M. (2005). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • [37] Skorokhod, A. V. (1961). Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6 264–274.
  • [38] Stevens, A. (2000). The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61 183–212.
  • [39] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, Berlin.
  • [40] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École d’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin.
  • [41] Velazquez, J. J. L. (2004). Point dynamics in a singular limit of the Keller–Segel model. I. Motion of the concentration regions. SIAM J. Appl. Math. 64 1198–1223.
  • [42] Velazquez, J. J. L. (2004). Point dynamics in a singular limit of the Keller–Segel model. II. Formation of the concentration regions. SIAM J. Appl. Math. 64 1224–1248.