Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.13117 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916214984409088 |
|---|---|
| author | Banerjee, Sayan Budhiraja, Amarjit Imon, Dilshad |
| author_facet | Banerjee, Sayan Budhiraja, Amarjit Imon, Dilshad |
| contents | We study a model for flocking given by a $n$-particle system under which each particle jumps forward by a random amount, independently sampled from a given distribution $θ$, with rate given by a non-increasing function $w$ of its signed distance from the system center of mass. This model was introduced in Balázs et. al. (2014) and some of its properties were studied for the case when $w$ is bounded. In the current work we are interested in the setting where $w$ is unbounded, and this feature results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We characterize the large $n$ limit (the so-called `fluid limit') of the empirical measure process associated with the system and prove a propagation of chaos result. Next, for the centered $n$-particle system, by constructing suitable Lyapunov functions, we establish 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, by establishing that all stationary solutions of the McKean-Vlasov equation must be the unique fixed point of the equation, we prove a propagation of chaos result at $t=\infty$ and establish convergence of the particle system, starting from stationarity, in the large $n$ limit, to a traveling wave solution of the McKean-Vlasov equation. The proof of this result may be of interest for other interacting particle systems where convexity properties or functional inequalities generally used for establishing such a result are not available. Our work answers several open problems posed in Balázs et. al. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13117 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flocking under Fast and Large Jumps: Stability, Chaos, and Traveling Waves Banerjee, Sayan Budhiraja, Amarjit Imon, Dilshad Probability We study a model for flocking given by a $n$-particle system under which each particle jumps forward by a random amount, independently sampled from a given distribution $θ$, with rate given by a non-increasing function $w$ of its signed distance from the system center of mass. This model was introduced in Balázs et. al. (2014) and some of its properties were studied for the case when $w$ is bounded. In the current work we are interested in the setting where $w$ is unbounded, and this feature results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We characterize the large $n$ limit (the so-called `fluid limit') of the empirical measure process associated with the system and prove a propagation of chaos result. Next, for the centered $n$-particle system, by constructing suitable Lyapunov functions, we establish 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, by establishing that all stationary solutions of the McKean-Vlasov equation must be the unique fixed point of the equation, we prove a propagation of chaos result at $t=\infty$ and establish convergence of the particle system, starting from stationarity, in the large $n$ limit, to a traveling wave solution of the McKean-Vlasov equation. The proof of this result may be of interest for other interacting particle systems where convexity properties or functional inequalities generally used for establishing such a result are not available. Our work answers several open problems posed in Balázs et. al. |
| title | Flocking under Fast and Large Jumps: Stability, Chaos, and Traveling Waves |
| topic | Probability |
| url | https://arxiv.org/abs/2404.13117 |