arXiv subjects

Aline Kurtzmann

Publications and source records attributed to Aline Kurtzmann.

9 recordsLinked to original sources

arXiv2025

Exit-problem for a class of non-Markov processes with path dependency

We study the exit-time of a self-interacting diffusion from an open domain $G \subset \mathbb{R}^d$. In particular, we consider the equation $d{X_t} = - \left( \nabla V(X_t) + \frac{1}{t}\int_0^t\nabla F (X_t - X_s)d{s} \right) d{t} + σd{W_t}.$ We are interested in the small-noise ($σ\to 0$) behaviour of the exit-time from the potentials' domain of attraction. In this work rather weak assumptions on the potentials $V$ and $F$, and on the domain $G$ are considered. In particular, we do not assume $V$ nor $F$ to be either convex or concave, which covers a wide range of self-attracting and self-repelling stochastic processes possibly moving in a complex multi-well landscape. The Large Deviation Principle for the Self-interacting diffusion with generalized initial conditions is established. The main result of the paper states that, under some assumptions on the potentials $V$ and $F$, and on the domain $G$, the Kramers' type law for the exit-time holds. Finally, we provide a result concerning the exit-location of the diffusion.

math.PR↗

arXiv2023

Self-interacting diffusions: long-time behaviour and exit-problem in the convex case

We study a class of time-inhomogeneous diffusion: the self-interacting one. We show a convergence result with a rate of convergence that does not depend on the diffusion coefficient. Finally, we establish a so-called Kramers' type law for the first exit-time of the process from domain of attractions when the landscapes are uniformly convex.

math.PR↗

arXiv2022

On the exit-problem for self-interacting diffusions

We study the exit-time from a domain of a self-interacting diffusion, where the Brownian motion is replaced by $σB_t$ for a constant $σ$. The first part of this work consists in showing that the rate of convergence (of the occupation measure of the self-interacting process toward some explicit Gibbs measure) previously obtained in \cite{kk-ejp} for a convex confinment potential $V$ and a convex interaction potential can be bounded uniformly with respect to $σ$. Then, we prove an Arrhenius-type law for the first exit-time from a domain (satisfying classical hypotheses of Freidlin-Wentzell theory).

math.PR↗

arXiv2016

On the Stability and the Exponential Concentration of Extended Kalman-Bucy filters

The exponential stability and the concentration properties of a class of extended Kalman-Bucy filters are analyzed. New estimation concentration inequalities around partially observed signals are derived in terms of the stability properties of the filters. These non asymptotic exponential inequalities allow to design confidence interval type estimates in terms of the filter forgetting properties with respect to erroneous initial conditions. For uniformly stable signals, we also provide explicit non-asymptotic estimates for the exponential forgetting rate of the filters and the associated stochastic Riccati equations w.r.t. Frobenius norms. These non asymptotic exponential concentration and quantitative stability estimates seem to be the first results of this type for this class of nonlinear filters. Our techniques combine $χ$-square concentration inequalities and Laplace estimates with spectral and random matrices theory, and the non asymptotic stability theory of quadratic type stochastic processes.

math.PR↗

arXiv2016

On the stability and the uniform propagation of chaos of a class of Extended Ensemble Kalman-Bucy filters

This article is concerned with the exponential stability and the uniform propagation of chaos properties of a class of Extended Ensemble Kalman-Bucy filters with respect to the time horizon. This class of nonlinear filters can be interpreted as the conditional expectations of nonlinear McKean Vlasov type diffusions with respect to the observation process. In contrast with more conventional Langevin nonlinear drift type processes, the mean field interaction is encapsulated in the covariance matrix of the diffusion. The main results discussed in the article are quantitative estimates of the exponential stability properties of these nonlinear diffusions. These stability properties are used to derive uniform and non asymptotic estimates of the propagation of chaos properties of Extended Ensemble Kalman filters, including exponential concentration inequalities. To our knowledge these results seem to be the first results of this type for this class of nonlinear ensemble type Kalman-Bucy filters.

math.PR↗

arXiv2015

A counterexample to the Cantelli conjecture through the Skorokhod embedding problem

In this paper, we construct a counterexample to a question by Cantelli, asking whether there exists a nonconstant positive measurable function $φ$ such that for i.i.d. r.v. $X,Y$ of law $\mathcal{N}(0,1)$, the r.v. $X+φ(X)\cdot Y$ is also Gaussian. This construction is made by finding an unusual solution to the Skorokhod embedding problem (showing that the corresponding Brownian transport, contrary to the Root barrier, is not unique). To find it, we establish some sufficient conditions for the continuity of the Root barrier function.

math.PR↗

arXiv2012

Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence

This paper deals with some self-interacting diffusions $(X_t,t\geq 0)$ living on $\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: \[\mathrm{d}X_t=\mathrm{d}B_t-g(t)\nabla V(X_t-\barμ_t)\,\mathrm{d}t,\] where $\barμ_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behaviour of $X$ and prove that it is strongly related to $g$. Actually, we show that $X$ is ergodic (in the limit quotient sense) if and only if $\barμ_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$.

math.PR↗

arXiv2010

Ergodicity of self-attracting motion

The aim of this paper is to study the asymptotic behaviour of a class of self- attracting motions on R^d . Using stochastic approximation methods, these processes have already been studied by Benaïm, Ledoux and Raimond (2002) in a compact setting. We also relate the asymptotic behaviour of the self-attracting Brownian motion to the McKean-Vlasov process that was studied, via the decrease of the free energy, by Carrillo, McCann and Villani (2003). Mixing these methods, we manage to obtain sufficient conditions for the (limit-quotient) ergodicity of the self-attracting diffusion, together with a speed of convergence.

math.PR↗

arXiv2008

Convergence in distribution of some particular self-interacting diffusions: the simulated annealing method

The present paper is concerned with some self-interacting diffusions $(X_t,t\geq 0)$ living on $\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t - g(t)\nabla V(X_t - \barμ_t) \mathrm{d}t$$ where $\barμ_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. The authors have still studied the ergodic behavior of $X$ and proved that it is strongly related to $g$. We go further and give necessary and sufficient conditions (for small $g$'s) in order that $X$ converges in probability to $X_\infty$ (which is related to the global minima of $V$).

math.PR↗