运动助力接触过程：泊松布朗运动上逾渗阈值以下的存活
Motion helps the contact process: survival below the percolation threshold on Poisson Brownian motions
AI总结：
该研究针对泊松布朗运动上的SIS接触过程，证明当恢复速率足够小时，在低于静态布尔模型临界强度的条件下，感染可存活并以正速度传播。
AI中文摘要：
我们考虑在d维欧氏空间中强度为η的独立布朗运动泊松系统上的SIS接触过程。当两个粒子的距离不超过2r时，它们处于接触状态。被感染粒子以速率λ∈(0,∞]将感染传播给与其接触的易感粒子，并以速率μ∈(0,∞]恢复，恢复后再次变为易感粒子。当λ=∞时，恢复仅在粒子孤立时发生。设η_c^B为静态布尔模型的临界强度，低于该强度时，接触图在每个固定时刻仅包含有限分量。我们证明，对于d≥2，在运动存在的情况下，当恢复速率μ足够小（取决于强度）时，感染在严格低于η_c^B的每个强度下均能存活并以正速度传播；即从单个被感染粒子开始，在时间t时，存在一个被感染粒子距离原点的量级为t。特别地，存在μ_†>0，使得对于每个μ<μ_†，有η_c(∞,μ)<η_c^B，且当μ↓0时，感染存活的强度集合扩展至全部(0,η_c^B)。我们还证明，在每个恢复速率下，临界密度均为正，其下界不依赖于感染速率；并且对于d≥2，在每个强度和每个有限感染速率下，当μ足够小时，感染均能存活并以正速度传播。
英文摘要：
We consider the SIS contact process on a Poisson system of independent Brownian motions in $\mathbb{R}^d$ of intensity $η$. Two particles are in contact when their distance is at most $2r$. An infected particle transmits the infection to a susceptible particle in contact with it at rate $λ\in(0,\infty]$, and recovers at rate $μ\in(0,\infty]$, after which it is susceptible again. When $λ=\infty$, recovery acts only while a particle is isolated. Let $η_c^{\mathrm{B}}$ be the critical intensity of the static Boolean model, below which the contact graph has only finite components at every fixed time. We show that for $d\ge2$, with motion, at every intensity strictly below $η_c^{\mathrm{B}}$, the infection survives and spreads at positive speed once the recovery rate $μ$ is small enough, depending on the intensity; that is, started from a single infected particle, at time $t$ there is an infected particle at distance of order $t$ from the origin. In particular there exists $μ_\dagger>0$ such that $η_c(\infty,μ)<η_c^{\mathrm{B}}$ for every $μ<μ_\dagger$, and as $μ\downarrow0$ the set of intensities at which the infection survives extends to all of $(0,η_c^{\mathrm{B}})$. We also show that the critical density is positive at every recovery rate, with a lower bound that does not depend on the infection rate, and that for $d\ge2$, at every intensity and every finite infection rate, the infection survives and spreads at positive speed once $μ$ is small enough.
