arXiv subjects

Jérémy Zurcher

Publications and source records attributed to Jérémy Zurcher.

4 recordsLinked to original sources

arXiv2026

Multidimensional Stein's method for asymptotic independence with invariant measures of diffusion

We derive a multidimensional Stein's method for asymptotic independence in the case of a general target $μ$ with a density, being invariant measure of a diffusion process. It allows us to give a general bound in Wasserstein distance between the law of a couple $(X, Y)$, where $X$ is a random variable, and $Y$ a random vector and $μ\otimes \mathrm{Law}(Y)$. We focus in particular in the case where $X$ and $Y$ are differentiable in the Malliavin sense, by being function of a finite number of stochastic Wiener integrals.

math.PR↗

arXiv2025

A Poisson-Alekseev-Gröbner formula through Malliavin calculus for Poisson random integrals

In this paper, we establish an Alekseev--Gröbner formula for stochastic differential equations (SDEs) driven by a Poisson random measure, which express the global error between a functional of two processes solution of SDEs started at the same initial condition, in terms of the infinitesimal error (i.e, the difference between the SDEs coefficients). In particular, we consider the situation where the flow process is only assumed to be jointly stochastically continuous with respect to space and time. Our proof relies on a new approach for the definition of the Skorohod--Poisson integral to treat the anticipating term appearing in the formula, based on a definition of the Malliavin derivative on a class of smooth random variables, instead of the more standard polynomial chaos approach.

math.PR↗

arXiv2024

The spatial average of solutions to SPDEs is asymptotically independent of the solution

Let $\left(u(t,x), t\geq 0, x\in \mathbb{R}^d\right)$ be the solution to the stochastic heat or wave equation driven by a Gaussian noise which is white in time and white or correlated with respect to the spatial variable. We consider the spatial average of the solution $F_{R}(t)= \frac{1}{σ_R}\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx$, where $σ^2_R= \mathbf{E} \left(\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx\right)^2$. It is known that, when $R$ goes to infinity, $F_R(t)$ converges in law to a standard Gaussian random variable $Z$. We show that the spatial average $F_R(t)$ is actually asymptotic independent by the solution itself, at any time and at any point in space, meaning that the random vector $(F_R(t), u(t, x_0))$ converges in distribution, as $R\to \infty$, to $(Z, u(t, x_0))$, where $Z$ is a standard normal random variable independent of $u(t, x_0)$. By using the Stein-Malliavin calculus, we also obtain the rate of convergence, under the Wasserstein distance, for this limit theorem.

math.PR↗

arXiv2023

Multidimensional Stein's method for Gamma approximation

Let F ($ν$) be the centered Gamma law with parameter $ν$ > 0 and let us denote by P Y the probability distribution of a random vector Y. We develop a multidimensional variant of the Stein's method for Gamma approximation that allows to obtain bounds for the second Wasserstein distance between the probability distribution of an arbitrary random vector (X, Y) in R x R n and the probability distribution F ($ν$) $\otimes$ P Y. In the case of random vectors with components in Wiener chaos, these bounds lead to some interesting criteria for the joint convergence of a sequence ((X n , Y n), n $\ge$ 1) to F ($ν$) $\otimes$ P Y , by assuming that (X n , n $\ge$ 1) converges in law, as n $\rightarrow$ $\infty$, to F ($ν$) and (Y n , n $\ge$ 1) converges in law, as n $\rightarrow$ $\infty$, to an arbitrary random vector Y. We illustrate our criteria by two concrete examples.

math.PR↗