arXiv subjects

Gerardo Barrera

Publications and source records attributed to Gerardo Barrera.

At least 19 recordsLinked to original sources

arXiv2026

Ergodicity and mixing for locally monotone stochastic evolution equations

We establish general quantitative conditions for stochastic evolution equations with locally monotone drift and degenerate additive Wiener noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated exponentially ergodic Markovian Feller semigroup. We prove improved moment estimates for the solutions and the $e$-property of the semigroup. Furthermore, we provide quantitative upper bounds for the $2$-Wasserstein $\varepsilon$-mixing times. Examples on possibly unbounded domains include the stochastic incompressible 2D Navier-Stokes equations, shear thickening stochastic power-law fluid equations, the stochastic heat equation, as well as, stochastic semilinear equations such as the 1D stochastic Burgers equation.

math.PR↗

arXiv2026

$χ^2$-cut-off phenomenon for Galerkin projections of Fokker-Planck equations with monomial potentials

In this manuscript, we establish the existence/non-existence of the cut-off phenomenon for the Langevin--Kolmogorov random dynamics with monomial convex potentials, possible singular, and driven by a Brownian motion with small strength. We consider a truncated $χ^2$-distance, that is, a distance based on Galerkin projections of the eigensystem, and show that not only a refined knowledge of the eigenvalues is needed but also a refined asymptotics of the growth for the eigenfunctions of the Fokker--Planck equations associated to the Langevin--Kolmogorov dynamics. In addition, this explicit analysis yields asymptotics of the mixing times and, in some regimes, information on the limiting profile, going beyond the product condition and the cut-off window alone.

math.PR↗

arXiv2026

Mixing times for the stochastic $p$-Laplace equation

We give an overview on existing quantitative results on long-time behavior of the stochastic $p$-Laplace equation with additive Wiener noise, $p>1$. We summarize the existing results in a table. We give explicit quantitative upper and lower estimates for the $\varepsilon$-mixing times of the stochastic $p$-Laplace equations for $p>1$. We summarize the mixing time asymptotics in a table.

math.PR↗

arXiv2026

Ergodicity bounds in the Sliced Wasserstein distance for Schur stable autoregressive processes

Explicit calculations in dimension one show for Schur stable autoregressive processes with standard Gaussian noise that the ergodic convergence in the Wasserstein-$2$ distance is essentially given by the sum of the mean, which decays exponentially, and the standard deviation, which decays with twice the speed. This paper starts by showing new upper and lower multivariate affine transport bounds for the Wasserstein-$r$ distance for $r$ greater and equal to $1$. These bounds allow to formulate a novel sufficient (non-Gaussian) ergodic interpolation condition for the mentioned mean-variance behavior to take place in case of more general Schur stable multivariate autoregressive processes. All ergodic estimates are non-asymptotic with completely explicit constants. The main applications are precise thermalization bounds for Schur stable $\mathsf{AR}(p)$ and $\mathsf{ARMA}(p,q)$ models in Wasserstein and Sliced Wasserstein distance. In the sequel we establish with the help of coupling techniques explicit upper and lower exponential bounds for more general multivariate Schur stable autoregressive processes. This includes parallel sampling and the convergence of the empiricial means. The utility of our results in particular for the Sliced Wasserstein distance are confirmed by multivariate numerical experiments.

math.PR↗

arXiv2025

Wasserstein error estimates between telegraph processes and Brownian motion

We provide non-asymptotic error bounds in the path Wasserstein distance with quadratic integral cost between suitable functionals of the telegraph process and the corresponding functional of Brownian motion with explicit diffusivity constant. These results cover, in particular, the well-known example of the exponential integral functional of the geometric Brownian motion. The non-asymptotic error bounds tend to zero in the so-called Kac regime. Moreover, the error bounds remain valid when the flip rate for the telegraph process is small. We assess the sharpness of the error bounds through numerical experiments.

math.PR↗

arXiv2025

Cut-off phenomenon and asymptotic mixing for multivariate general linear processes

The small noise cut-off phenomenon in continuous time and space has been studied in the recent literature for the linear and non-linear stable Langevin dynamics with additive Lévy drivers - understood as abrupt thermalization of the system along a particular time scale to its dynamical equilibrium - both for the total variation distance and the Wasserstein distance. The main result of this article establishes sufficient conditions for the window and profile cut-off phenomenon, which are flexible enough to cover the renormalized (non-Markovian) Ornstein--Uhlenbeck process driven by fractional Brownian motion and a large class of Gaussian and non-Gaussian, homogeneous and non-homogeneous drivers with (possible) finite second moments. The sufficient conditions are stated both for the total variation distance and the Wasserstein distance. Important examples are the multidimensional fractional Ornstein--Uhlenbeck process, the empirical sampling process of a fractional Ornstein--Uhlenbeck process, an Ornstein--Uhlenbeck processes driven by an Ornstein--Uhlenbeck process and the inhomogeneous Ornstein--Uhlenbeck process arising in simulated annealing.

math.PR↗

arXiv2025

Profile cut-off phenomenon for the ergodic Feller root process

The present manuscript is devoted to the study of the convergence to equilibrium as the noise intensity $\varepsilon>0$ tends to zero for ergodic random systems out of equilibrium of the type \begin{align*} \mathrm{d} X^{\varepsilon}_t(x) = (\mathfrak{b}-\mathfrak{a} X^{\varepsilon}_t(x))\mathrm{d} t+\varepsilon \sqrt{X^{\varepsilon}_t(x)}\mathrm{d} B_t, \quad X^{\varepsilon}_0(x) = x, \quad t\geqslant 0, \end{align*} where $x\geqslant 0$, $\mathfrak{a}>0$ and $\mathfrak{b}>0$ are constants, and $(B_t)_{t \geqslant 0}$ is a one dimensional standard Brownian motion. More precisely, we show the strongest notion of asymptotic profile cut-off phenomenon in the total variation distance and in the renormalized Wasserstein distance when $\varepsilon$ tends to zero with explicit cut-off time, explicit time window, and explicit profile function. In addition, asymptotics of the so-called mixing times are given explicitly.

math.PR↗

arXiv2024

The stability of the multivariate geometric Brownian motion as a bilinear matrix inequality problem

In this manuscript, we study the stability of the origin for the multivariate geometric Brownian motion. More precisely, under suitable sufficient conditions, we construct a Lyapunov function such that the origin of the multivariate geometric Brownian motion is globally asymptotically stable in probability. Moreover, we show that such conditions can be rewritten as a Bilinear Matrix Inequality (BMI) feasibility problem. We stress that no commutativity relations between the drift matrix and the noise dispersion matrices are assumed and therefore the so-called Magnus representation of the solution of the multivariate geometric Brownian motion is complicated. In addition, we exemplify our method in numerous specific models from the literature such as random linear oscillators, satellite dynamics, inertia systems, diagonal and non-diagonal noise systems, cancer self-remission and smoking.

math.PR↗

arXiv2024

Gradual convergence for Langevin dynamics on a degenerate potential

In this paper, we study an ordinary differential equation with a degenerate global attractor at the origin, to which we add a white noise with a small parameter that regulates its intensity. Under general conditions, for any fixed intensity, as time tends to infinity, the solution of this stochastic dynamics converges exponentially fast in total variation distance to a unique equilibrium distribution. We suitably accelerate the random dynamics and show that the preceding convergence is gradual, that is, the function that associates to each fixed $t\geq 0$ the total variation distance between the accelerated random dynamics at time $t$ and its equilibrium distribution converges, as the noise intensity tends to zero, to a decreasing function with values in $(0,1)$. Moreover, we prove that this limit function for each fixed $t \geq 0$ corresponds to the total variation distance between the marginal, at time $t$, of a stochastic differential equation that comes down from infinity and its corresponding equilibrium distribution. This completes the classification of all possible behaviors of the total variation distance between the time marginal of the aforementioned stochastic dynamics and its invariant measure for one dimensional well-behaved convex potentials. In addition, there is no cut-off phenomenon for this one-parameter family of random processes and asymptotics of the mixing times are derived.

math.PR↗

arXiv2024

Cutoff ergodicity bounds in Wasserstein distance for a viscous energy shell model with Lévy noise

This article establishes explicit non-asymptotic ergodic bounds in the renormalized Wasserstein-Kantorovich-Rubinstein (WKR) distance for a viscous energy shell lattice model of turbulence with random energy injection. The system under consideration is driven either by a Brownian motion, a symmetric $α$-stable Lévy process, a stationary Gaussian or $α$-stable Ornstein-Uhlenbeck process, or by a general Lévy process with second moments. The obtained non-asymptotic bounds establish asymptotically abrupt thermalization. The analysis is based on the explicit representation of the solution of the system in terms of convolutions of Bessel functions.

math-ph↗

arXiv2023

Egerváry's theorems for harmonic trinomials

In this manuscript, we study the arrangements of the roots in the complex plane for the lacunary harmonic polynomials called harmonic trinomials. We provide necessary and sufficient conditions so that two general harmonic trinomials have the same set of roots up to a rotation around the origin in the complex plane, a reflection over the real axis, or a composition of the previous both transformations. This extends the results of J. Egerváry 1930 for the setting of trinomials to the setting of harmonic trinomials.

math.CA↗

arXiv2023

Ergodicity bounds for stable Ornstein-Uhlenbeck systems in Wasserstein distance with applications to cutoff stability

This article establishes cutoff stability also known as abrupt thermalization for generic multidimensional Hurwitz stable Ornstein-Uhlenbeck systems with (possibly degenerate) Lévy noise at fixed noise intensity. The results are based on several ergodicity quantitative lower and upper bounds some of which make use of the recently established shift linearity property of the Wasserstein-Kantorovich-Rubinstein distance by the authors. It covers such irregular systems like Jacobi chains and more general networks of coupled harmonic oscillators with a heat bath (including Lévy excitations) at constant temperature on the outer edges and the so-called Brownian gyrator.

math.PR↗

arXiv2023

The stability region for Schur stable trinomials with general complex coefficients

In this paper, we characterize the stability region for trinomials of the form $f(ζ):=aζ^n + bζ^m +c$, $ζ\in \mathbb{C}$, where $a$, $b$ and $c$ are non-zero complex numbers and $n,m\in \mathbb{N}$ with $n>m$. More precisely, we provide necessary and sufficient conditions on the coefficients $a$, $b$ and $c$ in order that all the roots of the trinomial $f$ belongs to the open unit disc in the complex plane. The proof is based on Bohl's Theorem introduced in 1908.

math.CV↗

arXiv2022

The asymptotic distribution of the condition number for random circulant matrices

In this manuscript, we study the limiting distribution for the joint law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition. Under an appropriated normalization, the joint law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws. The latter implies that a normalized condition number converges in distribution to a Fréchet law as the dimension of the matrix increases.

math.PR↗

arXiv2022

Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process

In this manuscript, we provide a non-asymptotic process level control between the telegraph process and the Brownian motion with suitable diffusivity constant via a Wasserstein distance with quadratic average cost. In addition, we derive non-asymptotic estimates for the corresponding time average $p$-th moments. The proof relies on coupling techniques such as coin-flip coupling, synchronous coupling and the Komlós--Major--Tusnády coupling.

math.PR↗

arXiv2022

Salem-Zygmund Inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices

In this manuscript we give an extension of the classic Salem--Zygmund inequality for locally sub-Gaussian random variables. As an application, the concentration of the roots of a Kac polynomial is studied, which is the main contribution of this manuscript. More precisely, we assume the existence of the moment generating function for the iid random coefficients for the Kac polynomial and prove that there exists an annulus of width \[O(n^{-2}(\log n)^{-1/2-γ}), \quad γ>1/2\] around the unit circle that does not contain roots with high probability. As an another application, we show that the smallest singular value of a random circulant matrix is at least $n^{-ρ}$, $ρ\in(0,1/4)$ with probability $1-O(n^{-2ρ})$.

math.PR↗

arXiv2022

On the number of roots for harmonic trinomials

In this manuscript we study the counting problem for harmonic trinomials of the form $aζ^n+b\overlineζ^m+c$, where $n,m\in \mathbb{N}$, $n>m$, and $a$, $b$ and $c$ are non-zero complex numbers. As a consequence, we obtain the Fundamental Theorem of Algebra and the Wilmshurst conjecture for harmonic trinomials. The proof of the counting problem relies on the Bohl method introduced in Bohl (1908).

math.CV↗

arXiv2022

Limit behavior of the invariant measure for Langevin dynamics

In this manuscript, we consider the Langevin dynamics on $\mathbb{R}^d$ with an overdamped vector field and driven by multiplicative Brownian noise of small amplitude $\sqrtε$, $ε>0$. Under suitable assumptions on the vector field and the diffusion coefficient, it is well-known that it possesses a unique invariant probability measure $μ^ε$. As $ε$ tends to zero, we prove that the probability measure $ε^{d/2} μ^ε(\sqrtε\mathrm{d} x)$ converges in the $p$-Wasserstein distance for $p\in [1,2]$ to a Gaussian measure with zero-mean vector and non-degenerate covariance matrix which solves a Lyapunov matrix equation. Moreover, the error term is estimated. We emphasize that generically no explicit formula for $μ^ε$ can be found.

math.PR↗