We focus on a stochastic system that models a collection of $N$ identical oscillators with unidirectional coupling, perturbed by white noise. The unidirectional coupling breaks the symmetry of mutual dependence among the oscillators. It is shown that the associated random dynamical system $\phi$ admits a dynamical order, meaning that $\phi$ admits a simple asymptotic one-dimensional structure that is totally ordered with respect to the standard order in $\mathbb{R}^N$. Our approach takes a novel geometric perspective: we introduce a random dynamical system $\Phi$ on a smooth Riemannian manifold $M$, diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^{N-1}$, by ``wrapping''the random system $\phi$ from $\mathbb{R}^N$ onto $M$. By choosing an appropriate random cone field $\mathcal{C}_M$ on $M$, we show that $\Phi$ is a differentially positive random system on $M$, which is a random counterpart of the differentially positive systems introduced by Forni and Sepulchre. We further demonstrate that the traditional well-known horizontal curves can be identified as the conal curves on $M$, thereby providing a crucial tool for establishing the dynamical order of the random system $\phi$.
Let $M$ be a closed Riemannian surface, and let $f_1,\ldots,f_s$ be area-preserving $C^2$ diffeomorphisms with fixed positive probabilities such that the resulting random dynamical system is uniformly expanding on average. For each $n$, choose $\omega_1,\ldots,\omega_n$ independently according to these probabilities an...
We show that finite-order exponential mixing does not imply the central limit theorem for smooth diffeomorphisms. For every integer $R\geq2$, we construct a $\mathcal{C}^{\infty}$ diffeomorphism $T:M\to M$ of a compact smooth manifold $M$ and a $T$-invariant probability measure $\mu$ such that $(T,\mu)$ is exponentiall...
Bernat Bassols Cornudella, Matheus M. Castro· 0 citations
We study how stationary variance is distributed among the principal components of a randomly coupled linear system near instability. The coupling matrix has entries $N^{-1/2}\xi_{ij}$, where the $\xi_{ij}$ are independent copies of a bounded, centered real random variable of variance one. As $N\to\infty$ and the excess...
We study Gaussian fluctuations of differential observables of centered stationary random fields on the discrete $d$-dimensional torus of mesh $1/N$. Our main result is a Central Limit Theorem for the fields $f\mapsto N^{-d/2-m}\sum_x\phi_x^N(Pf)(x/N)$, where $P$ is a scalar homogeneous constant-coefficient differential...
Let $K$ be the non-singleton attractor of a finite iterated function system $\Phi=\{\phi_i(x)=a_i O_i x+t_i\}_{i=1}^m$ on $\mathbb{R}^d$, where $0<a_i<1$, $t_i\in\mathbb{R}^d$ and the $O_i$ are orthogonal transformations on $\mathbb{R}^d$. Suppose that $\Phi$ satisfies the open set condition, but not the strong separat...
We study quantitative propagation of chaos for mean-field forward-backward stochastic differential equations with interactions through the empirical measures of the forward states, backward values, and diagonal martingale integrands. Our main contribution is to introduce an intermediate-particle comparison method that...
Shu-Xian Gao, Ying Hu, Jia-Qiang Wen· 1 citation
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