论文标题
具有随机跳跃的扩散无序平均场模型的退火极限
Annealed limit for a diffusive disordered mean-field model with random jumps
论文作者
论文摘要
我们研究$ n- $粒子平均场系统的序列，每个系统都由$ n $简单的点过程$ z^{n，i} $在随机环境中驱动。 Each $Z^{N,i}$ has the same intensity $(f(X^N_{t-}))_t$ and at every jump time of $Z^{N,i},$ the process $X^N$ does a jump of height $U_i/\sqrt{N}$ where the $U_i$ are disordered centered random variables attached to each particle.我们证明了$ x^n $的分配收敛性到某个限制过程$ \ bar x $，它是用高斯变量给出的随机环境的SDE的，具有有限维分布的收敛速度。该高斯变量由CLT创建为$u_i。$的餐厅额的限制。 [Erny，Löcherbach和Loukianova（2022）]。
We study a sequence of $N-$particle mean-field systems, each driven by $N$ simple point processes $Z^{N,i}$ in a random environment. Each $Z^{N,i}$ has the same intensity $(f(X^N_{t-}))_t$ and at every jump time of $Z^{N,i},$ the process $X^N$ does a jump of height $U_i/\sqrt{N}$ where the $U_i$ are disordered centered random variables attached to each particle. We prove the convergence in distribution of $X^N$ to some limit process $\bar X$ that is solution to an SDE with a random environment given by a Gaussian variable, with a convergence speed for the finite-dimensional distributions. This Gaussian variable is created by a CLT as the limit of the patial sums of the $U_i.$ To prove this result, we use a coupling for the classical CLT relying on the result of [Komlós, Major and Tusnády (1976)], that allows to compare the conditional distributions of $X^N$ and $\bar X$ given the random environment, with the same Markovian technics as the ones used in [Erny, Löcherbach and Loukianova (2022)].
