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José R. León

Publications and source records attributed to José R. León.

11 recordsLinked to original sources
arXiv2026

Large Deviations and the Peano Phenomenon in Stochastic Differential Equations with Homogeneous Drift

We consider a diffusion equation in $\mathbb{R}^d$ with drift equal to the gradient of a homogeneous potential of degree $1+γ$, with $0<γ<1$, and local variance equal to $\varepsilon^2$ with $\varepsilon\to 0$. The associated deterministic system for $\varepsilon=0$ has a potential that is not a Lipschitz function at the origin. Therefore, an infinite number of solutions exist, known as the Peano phenomenon. In this work, we study first- and second-order large deviations for a noisy system, generalizing previous results for the specific potential $b(x)=x |x|^{γ-1}$. For the first-order large deviations, we recover the rate function from the well-known Freidlin-Wentzell work. For the second-order large deviation, we use a refinement of Carmona-Simon bounds for the eigenfunctions of a Schrödinger operator and prove that the exponential behavior of the process depends only on the ground state of such an operator. Moreover, a refined study of the ground state allows us to obtain the large deviation rate function explicitly and to deduce that the family of diffusions converges to the set of extreme solutions of the deterministic system.

math.PR↗
arXiv2023

On the number of roots of Sturm-Liouville random sums

We consider the number of roots of linear combinations of a system of $n$ orthogonal eigenfunctions of a Sturm-Liouville initial value problem with i.i.d. standard Gaussian coefficients. We prove that its distribution inherits the asymptotic behavior of the number of roots of Quall's random trigonometric polynomials. This result can be thought as a robustness result for the central limit theorem for the number of roots of Quall's random trigonometric polynomials in the sense that small uniform perturbations of sines and cosines do not change the limit distribution.

math.PR↗
arXiv2021

Studying the winding number for stationary Gaussian processes using real variables

We consider the winding number of planar stationary Gaussian processes defined on the line. Under mild conditions, we obtain the asymptotic variance and the Central Limit Theorem for the winding number as the time horizon tends to infinity. In the asymptotic regime, our discrete approach is equivalent to the continuous one studied previously in the literature and our main result extends the existing ones. Our model allows for a general dependence of the coordinates of the process and non-differentiability of one of them. Furthermore, beyond our general framework, we consider as examples an approximation to the winding number of a process whose coordinates are both non-differentiable and the winding number of a process which is not exactly stationary.

math.PR↗
arXiv2021

Central Limit Theorem for the volume of the zero set of Kostlan-Shub-Smale random polynomial systems

We state the Central Limit Theorem, as the degree goes to infinity, for the normalized volume of the zero set of a rectangular Kostlan-Shub-Smale random polynomial system. This paper is a continuation of {\it Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems} by the same authors in which the square case was considered. Our main tools are the Kac-Rice formula for the second moment of the volume of the zero set and an expansion of this random variable into the Itô-Wiener Chaos.

math.PR↗
arXiv2016

Estimation for stochastic damping Hamiltonian systems under partial observation. III. Diffusion term

This paper is the third part of our study started with Cattiaux, León and Prieur [Stochastic Process. Appl. 124 (2014) 1236-1260; ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359-384]. For some ergodic Hamiltonian systems, we obtained a central limit theorem for a nonparametric estimator of the invariant density [Stochastic Process. Appl. 124 (2014) 1236-1260] and of the drift term [ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359-384], under partial observation (only the positions are observed). Here, we obtain similarly a central limit theorem for a nonparametric estimator of the diffusion term.

math.PR↗
arXiv2015

CLT for the zeros of Classical Random Trigonometric Polynomials

We prove a Central Limit Theorem for the number of zeros of random trigonometric polynomials of the form $K^{-1/2}\sum_{n=1}^{K} a_n\cos(nt)$, being $(a_n)_n$ independent standard Gaussian random variables. In particular, we prove the conjecture by Farahmand, Granville & Wigman that the variance is equivalent to $V^2K$, $0<V^2<\infty$, as $K\to\infty$. % The case of stationary trigonometric polynomials was studied by Granville & Wigman and by Aza\"\is & León. Our approach is based on the Hermite/Wiener-Chaos decomposition for square-integrable functionals of a Gaussian process and on Rice Formula for zero counting.

math.PR↗
arXiv2011

Rice formulae and Gaussian waves

We use Rice formulae in order to compute the moments of some level functionals which are linked to problems in oceanography and optics: the number of specular points in one and two dimensions, the distribution of the normal angle of level curves and the number of dislocations in random wavefronts. We compute expectations and, in some cases, also second moments of such functionals. Moments of order greater than one are more involved, but one needs them whenever one wants to perform statistical inference on some parameters in the model or to test the model itself. In some cases, we are able to use these computations to obtain a central limit theorem.

math.PR↗
arXiv2009

Some applications of Rice formulas to waves

We use Rice's formulas in order to compute the moments of some level functionals which are linked to problems in oceanography and optics. For instance, we consider the number of specular points in one or two dimensions, the number of twinkles, the distribution of normal angle of level curves and the number or the length of dislocations in random wavefronts. We compute expectations and in some cases, also second moments of such functionals. Moments of order greater than one are more involved, but one needs them whenever one wants to perform statistical inference on some parameters in the model or to test the model itself. In some case we are able to use these computations to obtain a Central Limit Theorem.

math.PR↗
arXiv2008

Estimation in models driven by fractional Brownian motion

Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type \[X(t)=c+\int_0^tσ\bigl(X(u)\bigr)\mathrm {d}b_H(u)+\int _0^tμ\bigl(X(u)\bigr)\mathrm {d}u.\] In different particular models where $σ(x)=σ$ or $σ(x)=σx$ and $μ(x)=μ$ or $μ(x)=μx$, we propose a central limit theorem for estimators of $H$ and of $σ$ based on regression methods. Then we give tests of the hypothesis on $σ$ for these models. We also consider functional estimation on $σ(\cdot)$ in the above more general models based in the asymptotic behavior of functionals of the 2nd-order increments of the fBm.

math.PR↗
arXiv2006

On the second moment of the number of crossings by a stationary Gaussian process

Cramér and Leadbetter introduced in 1967 the sufficient condition \[\frac{r''(s)-r''(0)}{s}\in L^1([0,δ],dx),\qquad δ>0,\] to have a finite variance of the number of zeros of a centered stationary Gaussian process with twice differentiable covariance function $r$. This condition is known as the Geman condition, since Geman proved in 1972 that it was also a necessary condition. Up to now no such criterion was known for counts of crossings of a level other than the mean. This paper shows that the Geman condition is still sufficient and necessary to have a finite variance of the number of any fixed level crossings. For the generalization to the number of a curve crossings, a condition on the curve has to be added to the Geman condition.

math.PR↗