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Rank-based Particle Systems

Among interacting particle systems, a rank-based particle model is defined as a multi-dimensional Markov_process whose dynamics depend on the order in which the coordinates are ranked. Let $X_{i}(t)$ be the position of particle $\,i=1,\ldots ,n\,$ on the real line at time $\,t\geq 0\,$ , and denote by $\,\max _{1\leq i\leq n}X_{i}(t)=X_{(1)}(t)\geq X_{(2)}(t)\geq \ldots \geq X_{(n)}(t)=\min _{1\leq i\leq n}X_{i}(t)\,\,$ the descending order statistics of the positions. Ties are resolved lexicographically, or by randomization. The diffusion processes in the Atlas models are prototypical examples of rank-based particle systems. A general such system can be rewritten as

${\begin{aligned}dX_{i}(t)=b_{i}(F(X_{i}(t);\rho ^{n}(t)))dt+\sigma _{i}(F(X_{i}(t);\rho ^{n}(t)))dW_{i}(t)\,,\qquad i=1,\cdots ,n\,,\end{aligned}}$

where $\rho ^{n}(t)=\sum _{i=1}^{n}\delta (X_{i}(t))\in {\mathbf {M} }\,$ is the empirical measure of the particle system, $F(\cdot \,;\rho ^{n}(t))$ is the corresponding cumulative distribution function, $(W_{1}(\cdot ),\ldots ,W_{n}(\cdot ))$ is an $n$ -dimensional Brownian motion and $\,b_{i}:[0,1]\times {\mathbf {M} }\to {\mathbf {R} }\,$ , $\sigma _{i}:[0,1]\times {\mathbf {M} }\to (0,\infty )$ are given functions (Shkolnikov (2012)).

Collisions of Particles

Since the collisions of particles are incidents of rank change, it is significant to understand how and when the particles collide (Harris (1965)). Three independent Brownian particles on the real line (components of three-dimensional standard Brownian motion) never collide at the same time; however, there may be collisions of three or more particles with some positive probability at a finite time for general rank-based particle systems (Bass and Pardoux (1987), Ichiba and Karatzas (2010)). A Brownian motion in a Weyl chamber is an example of an interacting particle system, where the process terminates if two particles collide.

Long-term Behaviors

The long-term asymptotic behavior of rank-based particle systems can be understood through projections onto lower-dimensional manifolds. For example, the $n$ -dimensional rank-based particle process in the Atlas models is projected on the hyperplane $\,\{x\in {\mathbf {R} }^{n}:x_{1}+\cdots +x_{n}=0\}\,$ ; then the projected process is the centered process obtained by subtracting the average from each of the coordinates. Another example of projected processes is the gap process $\,(X_{(1)}(\cdot )-X_{(2)}(\cdot ),\ldots ,X_{(n-1)}(\cdot )-X_{(n)}(\cdot ))\,$ .

Under appropriate conditions, the projected process is positive recurrent in neighborhoods of the origin, and thus has a unique stationary distribution (e.g., Banner, Fernholz and Karatzas (2005), Ichiba et al. (2011), Chatterjee and Pal (2011) for the finite-dimensional case, and Pal and Pitman (2008), Chatterjee and Pal (2010) for the infinite-dimensional case). A similar quasi-invariant distribution is considered by Ruzmaikina and Aizenman (2005) and Arguin and Aizenman (2009), who connect the Ruelle Probability Cascades and hierarchically nested Poisson-Dirichlet processes with infinite interacting particle systems. The rate of convergence to the stationary distribution in the Atlas type models is studied by Jourdain and Malrieu (2008), Pal and Shkolnikov (2011), Ichiba, Pal and Shkolnikov (2013).

Infinite Dimension

Nonlinear evolution equations for porous media emerge as limits of finite-dimensional rank-based particle systems (Jourdain (2000 a,b), Shkolnikov (2011), Jourdain and Reygner (2012), Chatterjee and Pal (2010)). Under appropriate conditions, as the number of particles $\,n\,$ tends to infinity, the limiting particle density follows the McKean-Vlasov equation, and the corresponding cumulative distribution functions evolve according to the porous medium equation with convection (Shkolnikov (2012), Dembo, Shkolnikov, Varadhan and Zeitouni (2012)).

References

Arguin, L.P.; Aizenman, M. (2009). "On the structure of quasi-stationary competing particles systems". Annals of Probability. 37: 1080–1113.

Banner, A.; Fernholz, E.R.; Karatzas, I. (2005). "Atlas models of equity markets". Annals of Applied Probability. 15: 2296–2330.

Bass, R.; Pardoux, E. (1987). "Uniqueness for diffusions with piecewise constant coefficients". Probability Theory and Related Fields. 76: 557–572.

Chatterjee, S.; Pal, S. (2010). "A phase transition behavior for Brownian motions interacting through their ranks". Probability Theory and Related Fields. 147: 123–159.

Chatterjee, S.; Pal, S. (2011). "A combinatorial analysis of interacting diffusions". Journal of Theoretical Probability. 24: 939–968.

Dembo, A.; Shkolnikov, M.; Varadhan, S.R. Srinivasa; Zeitouni, O. (2012). "Large deviations for diffusions interacting through their ranks". arXiv:1211.5223.

Harris, T. E. (1965). "Diffusion with ``collisions between particles". Journal of Applied Probability. 2: 323–338.

Ichiba, T.; Karatzas, I. (2010). "Collisions of Brownian particles". Annals of Applied Probability. 20: 951–977.

Ichiba, T.; Papathanakos, V.,; Banner, A.D.; Karatzas, I.; Fernholz, E.R. (2011). "Hybrid Atlas Models". Annals of Applied Probability. 21: 609–644.{{cite journal}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)

Ichiba, T.; Pal, S.; Shkolnikov, M. (2013). "Convergence rates for rank-based models with applications to portfolio theory". Probability Theory and Related Fields. 156: 415–448.

Jourdain, B.; Malrieu, F. (2008). "Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF". Annals of Applied Probability. 18: 1706–1736.

Jourdain, B.; Reygner, J. (2012). "Propagation of chaos for rank-based interacting diffusions and long time behavior of a scalar quasilinear parabolic equation". arXiv:1211.4818.

Pal, S.; Pitman, J. (2008). "One dimensional Brownian particle systems with rank-dependent drifts". Annals of Applied Probability. 18: 2179-2207.

Pal, S.; Shkolnikov, M. (2011). "Concentration of measure for systems of Brownian particles interacting through their ranks". arXiv:1011.2443.

Ruzmaikina, A.; Aizenman, M. (2005). "Characterization of invariant measures at the leading edge for competing particle systems". Annals of Probability. 33: 82–113.

Shkolnikov, M. (2012). "Large systems of diffusions interacting through their ranks". Stochastic Processes and their Applications. 122: 1730–1747.

Shkolnikov, M. (2011). "Competing particle systems evolving by interacting Lévy processes". Annals of Applied Probability. 21: 1911–1932.