We study a class of diffusion processes on $\mathbb{R}^n$ interacting through a symmetric matrix $X\in\mathbb{R}^{n\times n}$. When eigenvectors of $X$ are Haar-uniform on the orthogonal group, we derive a dynamical mean-field limit for the empirical law of sample paths, extending the classical Sompolinsky--Zippelius characterization for $X\sim\mathrm{GOE}$. The limit takes the form of a generalized Langevin equation with correlated Gaussian noise and memory, whose correlation and response kernels relate to those of the original dynamics through convolution equations involving the free cumulants of the eigenvalue distribution of $X$. For the overdamped Langevin diffusion associated with $\mu(\boldsymbol{\theta})\propto \exp\!\big(\frac12\boldsymbol{\theta}^{\top}X\boldsymbol{\theta}\big)\prod_{i=1}^n\nu(\mathrm{d}\theta_i)$, we analyze the mean-field limit under a rapid-mixing assumption. The correlation and response kernels admit time-translation-invariant approximants satisfying a fluctuation-dissipation relation. The generalized Langevin equation admits a Markovian approximation coupled to an auxiliary multivariate OU process and converges to a replica-symmetric prediction for the empirical coordinate law under $\mu$. This auxiliary correlation structure is characterized through the infinitesimal generator of a Markov semigroup for the lifted path-history process. Consequently, the free energy converges to a replica-symmetric limit under an explicit high-temperature condition, which for an Ising model is $\|X\|_{\mathrm{op}}<1/2$. By recent dynamical universality results, the same free-energy characterization holds for deterministic models without random disorder when $X$ satisfies a set of deterministic delocalization conditions.
We prove a canonical (or fixed-population) local equilibrium theorem for the symmetric simple exclusion process on the discrete torus $\mathbb T_N^D$, $D\ge2$, at particle densities bounded away from $0$ and $1$, uniformly over all deterministic initial configurations with the prescribed particle number. At times \[ t_N(s)=\frac{\log(N^D)+s}{2\gamma_N}, \qquad \gamma_N=2-2\cos\left(\frac{2\pi}{N}\right), \] the Radon--Nikodym density of the process relative to equilibrium converges in $L^2$ to a canonical exponential tilt generated by the unique small mean-zero calibration field whose one-site marginals match the evolving one-particle heat profile. Consequently, whenever the profile coordinate converges, the corresponding total variation profile is a Gaussian shift. More precisely, for a deterministic initial sequence $(S_N)$, if the covariance-normalized squared amplitude $\mathfrak q_N^{S_N}(s)$ converges to a scalar $\mathfrak q$ at a fixed $s$, then the distance to stationarity converges to $ 2\Phi\!\left({\sqrt{\mathfrak q}}/2\right)-1 $, where $\Phi$ is the standard normal distribution function. The profile coordinate is asymptotically determined by the one-particle eigenspace corresponding to the smallest nonzero eigenvalue. The proof combines a calibrated canonical comparison, a fixed-degree comparison between independent and exclusion dynamics, and all-degree control obtained from pair energy estimates and preservation of the Strong--Rayleigh property.
We study the stationary focusing nonlinear Schr\"odinger equation on the product $\mathbb{R}^N \times \mathcal{G}$ of the Euclidean space with a compact metric graph, in the mass-constrained variational setting. Such a product is a hybrid structure of a new type: all its faces are $(N+1)$-dimensional and are glued along interfaces of codimension one, so that the energy space is a genuine Sobolev space, while both the metric and the topology of the graph enter the variational problem. We first develop the functional framework, giving two equivalent descriptions of $H^1(\mathbb{R}^N \times \mathcal{G})$, introducing partial rearrangements in each of the two variables together with the corresponding P\'olya--Szeg\H{o} inequalities, and proving Gagliardo--Nirenberg inequalities in a localized form, with a comparison of the optimal constants with those of $\mathbb{R}^{N+1}$ and of the half-space. We then study the mass-constrained problem. Ground states exist for every mass when $2<p<2_*:=2+4/(N+1)$. At the critical exponent $p=2_*$, they exist below a graph-dependent threshold lying between one half of the Euclidean critical mass and the full Euclidean critical mass. The latter value is attained when the graph admits a cycle covering, whereas the threshold is exactly halved in the presence of a terminal edge. In both cases, the threshold is sharp. For $2_*<p<2+4/N$ global minimizers do not exist, but we prove the existence of local minimizers below a further mass threshold, for which we give an explicit lower bound. Finally, we describe the dimensional crossover: below a critical mass the minimizers do not depend on the graph variable, and we characterize the threshold below which the semi-trivial solution is a local minimizer in terms of the first nonzero eigenvalue of the Kirchhoff Laplacian on $\mathcal{G}$.
Nicola Soave, G. Verzini, Lorenzo Villata· 0 citations
We consider the linear stochastic fractional diffusion equation \begin{equation*} \partial^{\beta} u(t,x)=-\left(-\Delta\right)^{\alpha/2}u(t,x) +I_t^{\gamma}\bigl[\dot W(t,x)\bigr], \qquad t>0,\quad x\in\mathbb R^d, \end{equation*} with zero initial conditions, where $\alpha>0$, $\beta\in(0,2)$, and $\gamma\ge0$. The driving noise $\dot W$ is a centered Gaussian generalized field that is fractional in time and has Riesz-type spatial covariance. For each fixed $x\in\mathbb R^d$, we establish a Khinchin-type law of the iterated logarithm at time zero for the temporal process $t\mapsto u(t,x)$. Under the additional conditions $0\le\gamma<1$ and $\beta+\gamma<2+H$, we also prove the corresponding Chung-type law. The proofs rely on a harmonizable representation, sharp frequency-truncation estimates, an exact small-ball asymptotic, and a localization argument. These results extend the initial-time laws of the iterated logarithm for stochastic heat equations to a broad class of time-fractional stochastic diffusion equations.
In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{\rho(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random. We assume that for each fixed realization of the environment, $\rho$ and $a$ are continuous and locally exponentially integrable, and that the support of the associated intrinsic coordinates is compact and non-collapsing. This framework includes the extensively studied Brox diffusion $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion representing the environment. The It\^o--McKean representation of the diffusions and the estimates of the first exit probabilities derived via Moser iteration play a crucial role.
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 $\phi$.
In this paper, we study the scaling limit of the pinning model in correlated Gaussian environment. The tail probability of the underlying renewal process of the model has a polynomial decay with exponent $\alpha>0$. The covariance of the Gaussian environment $\{\omega_n\}_{n\in\mathbb N}$ is given by $\text{Cov}_{\mathbb P}(\omega_n,\omega_m)\sim |n-m|^{2H-2}$ with $H\in(0,1)$. Assuming $\alpha\in(0,\frac12]$, $H\in(\frac12,1)$ and $\alpha+2H>2$, we show that the partition function of the disordered pinning model, under the appropriate scaling, converges in distribution to the $L^1$-solution of the fractional stochastic heat equation driven by Gaussian noise correlated in time and localized at the origin. In particular, it is known that the solution is not $L^2$-integrable when $\alpha<\frac12$.
Jian Song, Meng Wang, Ran Wei· 0 citations
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