This thesis studies asymptotic and ergodic properties of some flocking-type interacting particle systems. The systems we investigate include one with (global) mean-field interactions between the particles, while the other evolves through local interactions between neighbors. We are especially interested in how random interactions combine to generate complex collective behavior, the central theme being to establish rigorous limit theorems- such as hydrodynamic limits, ergodicity, mixing times, central limit theorems and identifying traveling wave phenomena. In the first two chapters, we study a pure-jump n-particle system with attractive mean field interactions under which each particle jumps forward by a random amount, independently sampled from a given distribution θ, at exponentially distributed times with rate given by a function w of its signed distance from the system center of mass. The function w is taken to be non-increasing which leads to a ‘flocking’ behavior: the particles below the center of mass jump forward at a higher rate than those above it. This model was introduced in (Bal´azs et al., 2014) and some of its properties were studied for the case when w is bounded. In our work, we are interested in the setting where w is unbounded, and this feature, together with the mild integrability we impose on the jump sizes, results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We identify natural conditions under which the system is well-posed and study the large n limit(the so-called ‘fluid limit’) of the empirical measure process associated with the system. Byestablishing well-posedness of the associated McKean-Vlasov equation we characterize the fluidlimit of the particle system and prove a propagation of chaos result. Next, for the centered n-particle system, by constructing suitable Lyapunov functions, weestablish the existence and uniqueness of stationary distributions and study their tail properties. In the special case where w is an exponential function and θ is an exponential distribution, byestablishing that all stationary solutions of the McKean-Vlasov equation must be the uniquefixed point of the equation, we prove a propagation of chaos result at t = ∞ and establishconvergence of the particle system, starting from stationarity, in the large n limit, to a travelingwave solution of the McKean-Vlasov equation. The proof of this result may be of interest forother interacting particle systems where convexity properties or functional inequalities generallyused for establishing such a result are not available. Our work answers several open problemsposed in (Bal´azs et al., 2014). In the third chapter, we introduce and analyze a class of interacting particle systems on thereal line that combine features of the stochastic rat race and (deterministic) follow-the-leadermodels. The particle system evolves as a continuous-time pure jump process: the leadingparticle moves independently, at Exponential jump times, with constant jump rate and iid jumpsizes distributed according to a law θ, while each of the remaining particles jumps forward, atExponential times, at rate equal to its distance from the particle immediately ahead, with jumpsizes drawn uniformly from the corresponding gap. The dynamics thus encode competitionfor leadership together with distance-dependent stochastic interactions. Our main focus is theassociated gap process, representing the vector of inter-particle distances. We establish theexistence of a unique stationary distribution for the gap process and prove uniform geometricergodicity. Further, when the leader’s jump sizes follow an Exponential distribution, we identifythe stationary law explicitly as a product of independent Exponential laws. For this case, wealso derive bounds on the mixing time, showing that it scales between Θ(n) and O(n(logn)2)for an n-particle system. As an application of the mixing time results we establish a functionallimit theorem that characterizes fluctuations of particle states at large time, under a suitablespatial and temporal scaling and large particle limit. Finally, when the leader’s jumps haveheavy but integrable tails, we show that each gap has at least one additional finite momentunder stationarity than that of the leader’s jump size distribution. Together, these resultsprovide a comprehensive analysis of stability and convergence to equilibrium in an interactingjump system with state-dependent rates and jump sizes, and local interactions. The modeloffers a tractable setting for exploring ergodicity, explicit invariant laws, and mixing behavior innon-diffusive particle systems.
Dilshad Imon (Fri,) studied this question.