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Christian Olivera

Publications and source records attributed to Christian Olivera.

At least 19 recordsLinked to original sources
arXiv2026

Parameter Estimation of the Stochastic Allen--Cahn Equation via variations

This paper addresses statistical inference for stochastic partial differential equations. We study the stochastic Allen-Cahn equation driven by space-time white noise and analyze its mild solution. Our main focus is the asymptotic behavior of the spatial quadratic variation of the solution, for which we establish the exact limiting value. As an application, we develop parameter estimation procedures based on these asymptotic results. We prove that the unique solution can be decomposed as u = X + Y where X denotes the solution of the linear stochastic heat equation and Y accounts for the nonlinear effects. Exploiting a detailed analysis of the heat kernel and its scaling behavior, we derive H\"older continuity properties of Y in both spatial and temporal variables, showing that Y exhibits substantially higher regularity than X. This decomposition and the resulting regularity estimates are key ingredients in the development of parameter estimation procedures based on the asymptotic behavior of quadratic variations of the solution.

math.PR↗
arXiv2026

Quantitative approximation of the Vlasov(-Fokker-Planck)-Navier-Stokes system by stochastic particle systems

This paper is concerned with a fluid-particle system given by the incompressible Navier-Stokes equations coupled with the Vlasov(-Fokker-Planck) equation through a drag force. Such a model arises naturally in the study of aerosols, sprays, and more generally two-phase flows. In dimensions $d\in \{2,3\}$, we establish a rate of convergence for a system of $N$ interacting stochastic particles coupled with a fluid, towards the Vlasov(-Fokker-Planck)-Navier-Stokes system, as $N\to \infty$. The case of particles with a noise that vanishes as $N\to \infty$ is considered and leads specifically to the Vlasov-Navier-Stokes system. More precisely, we prove that the empirical measure associated with the particle system converges to the Vlasov(-Fokker-Planck) component, while the fluid velocity converges to the Navier-Stokes component of the coupled system. The proofs combine stochastic calculus and PDE techniques to establish energy estimates and commutator estimates for both the discrete and continuous systems.

math.PR↗
arXiv2025

Pathwise quantitative particle approximation of nonlinear stochastic Fokker-Planck equations via relative entropy

We derive non-linear stochastic Fokker-Planck equation from stochastic systems particles with individual and environmental noise via relative entropy method, with pathwise quantitative bounds. Moreover, we prove the existence of a unique strong solution to the associated Fokker-Planck equation. Our proof is based on tools from PDE analysis, stochastic analysis, functional inequalities, and also we use the dissipation of entropy which provides some bound on the Fisher information of the particle system. The approach applies to repulsive and attractive kernels.

math.PR↗
arXiv2025

Temporal quadratic and higher order variation for the nonlinear stochastic heat equation and applications to parameter estimation

We consider the stochastic heat equation which includes a fractional power of the Laplacian of order $\alpha \in (1, 2]$ and it is driven by a nonlinear space-time Gaussian white noise. We study two types of power variations for the solution to this equation: the renormalized quadratic variation and the power variation of order $\frac{2\alpha}{\alpha -1}$, both over an equidistant partition of the unit interval. We prove that these two sequences admit nontrivial limits when the mesh of the partition goes to zero. We apply these results to identify certain parameters of the stochastic heat equation.

math.PR↗
arXiv2024

Quantitative particle approximation of nonlinear stochastic Fokker-Planck equations with singular kernel

We derive quantitative estimates for large stochastic systems of interacting particles perturbed by both idiosyncratic and environmental noises, as well as singular kernels. We prove that the (mollified) empirical process converges to the solution of the nonlinear stochastic Fokker-Planck equation. The proof is based on It\^o's formula for $H_{q}^{1}$-valued process, commutator estimates, and some estimations for the regularization of the empirical measure. Moreover, we show that the aforementioned equation admits a unique strong solution in the probabilistic sense. The approach applies to repulsive and attractive kernels.

math.PR↗
arXiv2024

An Eulerian-Lagrangian Formulation of the Compressible Euler Equations with Vacuum

In this paper, we present a novel Eulerian-Lagrangian formulation for the compressible isentropic Euler equations with vaccum. Using the developed Lagrangian flow map formulation, we show a short-time solution for a general pressure law. A particularly appealing feature of the approach used, it is well defined in the presence of vacuum, namely for compactly supported initial data which constitute an important problem in gas dynamics. Moreover, it does so without relying on any special symmetrization. While analogous results are well understood for incompressible fluids, the compressible setting, particularly in the presence of vacuum, remained open.

math.AP↗
arXiv2024

Microscopic derivation of non-local models with anomalous diffusions from stochastic particle systems

This paper considers a large class of nonlinear integro-differential scalar equations which involve an anomalous diffusion (e.g. driven by a fractional Laplacian) and a non-local singular convolution kernel. Each of those singular equations is obtained as the macroscopic limit of an interacting particle system modeled as N coupled stochastic differential equations driven by L\'evy processes. In particular we derive quantitative estimates between the microscopic empirical measure of the particle system and the solution to the limit equation in some non-homogeneous Sobolev space. Our result only requires very weak regularity on the interaction kernel, therefore it includes numerous applications, e.g.: the 2d turbulence model (including the quasi-geostrophic equation) in sub-critical regime, the 2d generalized Navier-Stokes equation, the fractional Keller-Segel equation in any dimension, and the fractal Burgers equation.

math.PR↗
arXiv2024

Convergence Rate for Moderate Interaction particles and Application to Mean Field Games

We study two interacting particle systems, both modeled as a system of $N$ stochastic differential equations driven by Brownian motions with singular kernels and moderate interaction. We show a quantitative result where the convergence rate depends on the moderate scaling parameter, the regularity of the solution of the limit equation and the dimension. Our approach is based on the techniques of stochastic calculus, some properties of Besov and Triebel-Lizorkin space, and the semigroup approach introduced in [9].

math.PR↗
arXiv2024

From particle systems to the stochastic compressible Navier-Stokes equations of a barotropic fluid

We propose a mathematical derivation of stochastic compressible Navier-Stokes equation. We consider many-particle systems with a Hamiltonian dynamics supplemented by a friction term and environmental noise. Both the interaction potential and the additional friction force are supposed to be long range in comparison with the typical distance between neighboring particles. It is shown that the empirical measures associated to the position and velocity of the system converge to the solutions of the stochastic compressible Navier-Stokes equations of a barotropic fluid. Moreover, we quantify the distance between particles and the limit in suitable in Besov and Triebel-Lizorkin spaces.

math.AP↗
arXiv2023

From Hamiltonian Systems to Compressible Euler Equation driven by additive H\"older noise

We derive stochastic compressible Euler Equation from a Hamiltonian microscopic dynamics. We consider systems of interacting particles with H\"older noise and potential whose range is large in comparison with the typical distance between neighbouring particles. It is shown that the empirical measures associated to the position and velocity of the system converge to the solutions of compressible Euler equations driven by additive H\"older path(noise), in the limit as the particle number tends to infinity, for a suitable scaling of the interactions. Furthermore, explicit rates for the convergence are obtained in Besov and Triebel-Lizorkin spaces. Our proof is based on the It\^o-Wentzell-Kunita formula for Young integral.

math.AP↗
arXiv2023

Euler-Lagrangian approach to stochastic Euler equations in Sobolev Spaces

The purpose of this paper is to establish the equivalence between Lagrangian and classical formulations for the stochastic incompressible Euler equations, the proof is based in Ito-Wentzell-Kunita formula and stochastic analysis techniques. Moreover, we prove a local existence result for the Lagrangian formulation in suitable Sobolev Spaces.

math.AP↗
arXiv2023

From Stochastic Hamiltonian Systems to Stochastic Compressible Euler Equation

We study a stochastic Hamiltonian system of $N$ particles with many particles interacting through a potential whose range is large in comparison with the typical distance between neighbouring particles. It is shown that the empirical measures associated to the position and velocity of the system converge to the solutions of stochastic compressible Euler equations in the limit as the particle number tends to infinity. Moreover, we quantify the distance between particles and the limit in suitable Sobolev norm.

math.AP↗
arXiv2021

Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel

In this work, we study the convergence of the empirical measure of moderately interacting particle systems with singular interaction kernels. First, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting nonlinear Fokker-Planck equation. Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos). Our results only require very weak regularity on the interaction kernel, including the Biot-Savart kernel, and attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. For some of these important examples, this is the first time that a quantitative approximation of the PDE is obtained by means of a stochastic particle system. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. The proofs are based on a semigroup approach combined with a fine analysis of the regularity of infinite-dimensional stochastic convolution integrals.

math.PR↗
arXiv2021

Stochastic transport equations with unbounded divergence

We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients and unbounded divergence. In the first result we assume the drift is $L^{2}([0,T] \times \R^{d})\cap L^{\infty}([0,T] \times \R^{d})$ and the divergence is the locally integrable. In the second result we show that the smoothing acts as a selection criterion when the drift is in $L^{2}([0,T] \times \R^{d})\cap L^{\infty}([0,T] \times \R^{d})$ without any condition on the divergence.

math.AP↗
arXiv2020

Stochastic transport equation in bounded domains

This paper is concerned with the initial-boundary value problem \; for stochastic transport equations in bounded domains. For a given stochastic perturbation of the drift vector field, we prove existence and uniqueness of weak solutions with non-regular coefficients. The existence of weak solutions relies strongly on the existence of strong stochastic trace, and also on the passage from the Stratonovich into Itô's formulation for bounded domains.Moreover, the uniqueness of weak solutions is obtained under suitable conditions.

math.AP↗
arXiv2020

Quantitative approximation of the Burgers and Keller-Segel equations by moderately interacting particles

In this work we obtain rates of convergence for two moderately interacting stochastic particle systems with singular kernels associated to the viscous Burgers and Keller-Segel equations. The main novelty of this work is to consider a non-locally integrable kernel. Namely for the viscous Burgers equation in $\mathbb{R}$, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some Bessel space with a rate of convergence of order $N^{-1/6}$, on any time interval. With the same rate, convergence also holds for the genuine empirical measure in Wasserstein distance, and at the level of the trajectories of the particles with the standard coupling to McKean-Vlasov particles. In the case of the Keller-Segel equation on a $d$-dimensional torus, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some $L^q$ space with a rate of order $N^{-\frac{1}{2(d+1)}}$. The result holds up to the maximal existence time of the PDE, for any value of the chemo-attractant sensitivity $\chi$.

math.PR↗