Related papers: The strong data processing inequality under the he…
We investigate percolation, a probabilistic model for continuous phase transition (CPT), on square and weighted planar stochastic lattices. In its thermal counterpart, entropy is minimally low where order parameter (OP) is maximally high…
This paper provides a precise error analysis for the maximum likelihood estimate $\hat{a}_{\text{ML}}(u_1^n)$ of the parameter $a$ given samples $u_1^n = (u_1, \ldots, u_n)'$ drawn from a nonstationary Gauss-Markov process $U_i = a U_{i-1}…
We consider correlated random variables $X_1,\dots,X_n$ taking values in $\{0,1\}$ such that, for any permutation $\pi$ of $\{1,\dots,n\}$, the random vectors $(X_1,\dots,X_n)$ and $(X_{\pi(1)},\dots,X_{\pi(n)})$ have the same distribution.…
Velocity and temperature distributions are both crucial for modeling compressible wall-bounded turbulent flows. The compressible law of the wall for velocity has been extensively examined through velocity transformations. However, a…
We present an extensive numerical study of the time irreversibility of the dynamics of heavy inertial particles in three-dimensional, statistically homogeneous and isotropic turbulent flows. We show that the probability density function…
Score estimation is the backbone of score-based generative models (SGMs), especially denoising diffusion probabilistic models (DDPMs). A key result in this area shows that with accurate score estimates, SGMs can efficiently generate samples…
We establish a strong Gaussian approximation for high-dimensional non-degenerate U-statistics with diverging dimension. Under mild assumptions, we construct, on a sufficiently rich probability space, a Gaussian process that uniformly…
This work explores the theoretical and practical foundations of denoising diffusion probabilistic models (DDPMs) and score-based generative models, which leverage stochastic processes and Brownian motion to model complex data distributions.…
We consider a diffusion process under a local weak H\"{o}rmander condition on the coefficients. We find Gaussian estimates for the density in short time and exponential lower and upper bounds for the probability that the diffusion remains…
This paper provides an elementary, self-contained analysis of diffusion-based sampling methods for generative modeling. In contrast to existing approaches that rely on continuous-time processes and then discretize, our treatment works…
In this paper we study the Heat Flow on Metric Random Walk Spaces, which unifies into a broad framework the heat flow on locally finite weighted connected graphs, the heat flow determined by finite Markov chains and some nonlocal evolution…
This work presents an upper-bound to value that the Kullback-Leibler (KL) divergence can reach for a class of probability distributions called quantum distributions (QD). The aim is to find a distribution $U$ which maximizes the KL…
We establish Gaussian-type upper bounds on the heat kernel for a continuous-time random walk on a graph with unbounded weights under an ergodicity assumption. For the proof we use Davies' perturbation method, where we show a maximal…
The Kullback-Leibler (KL) divergence is a foundational measure for comparing probability distributions. Yet in multivariate settings, its single value often obscures the underlying reasons for divergence, conflating mismatches in individual…
Consider a random walk $S_n=\sum_{i=1}^n X_i$ with independent and identically distributed real-valued increments with zero mean, finite variance and moment of order $2 + \delta$ for some $\delta>0$. For any starting point $x\in \mathbb R$,…
The first purpose of this note is to provide a proof of the usual square function estimate on Lp (?). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on Gaussian bounds…
We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of…
We study the canonical heat flow $(\mathsf{H}_t)_{t\geq 0}$ on the cotangent module $L^2(T^*M)$ over an $\mathrm{RCD}(K,\infty)$ space $(M,\mathsf{d},\mathfrak{m})$, $K\in\boldsymbol{\mathrm{R}}$. We show Hess-Schrader-Uhlenbrock's…
We employ novel digital Fresnel reflection holography to capture the 3D flows within the viscous sublayer of a smooth-wall turbulent channel flow at Re{\tau}=400. The measurements reveal unsteady and diverse flow patterns in the sublayer…
In this paper, we introduce a new way to estimate the scaling parameter of a self-similar process by considering the maximum probability density function (pdf) of tis increments. We prove this for $H$-self-similar processes in general and…