Large Deviations and Exit-times for reflected McKean-Vlasov equations with self-stabilizing terms and superlinear drifts

We study a class of reflected McKean-Vlasov diffusions over a convex domain with self-stabilizing coefficients. This includes coefficients that do not satisfy the classical Wasserstein Lipschitz condition. Further, the process is constrained to a (not necessarily bounded) convex domain by a local time on the boundary. These equations include the subclass of reflected self-stabilizing diffusions that drift towards their mean via a convolution of the solution law with a stabilizing potential. Firstly, we establish existence and uniqueness results for this class and address the propagation of chaos. We work with a broad class of coefficients, including drift terms that are locally Lipschitz in spatial and measure variables. However, we do not rely on the boundedness of the domain or the coefficients to account for these non-linearities and instead use the self-stabilizing properties. We prove a Freidlin-Wentzell type Large Deviations Principle and an Eyring-Kramer's law for the exit-time from subdomains contained in the interior of the reflecting domain.Comment: 41 page

Extraction d'un noyau non stationnaire de processus gaussien à l'aide des ondelettes

International audienceIn [1] Durrande et al. separate a classical gaussian process kernel (a Matern) into two sub kernels : one containing a random periodic component and the other being a residual kernel. This decomposition was inspired by Fourier serie decomposition and enable, by comparison of the two sub kernels, an assessment of how much periodicity our data have. Our goal is to show an extension of this work by using wavelets, providing us with a non stationary kernel. The end goal is to both extract and model non stationnary component of a noisy signal to analyse or to do simulations.Dans [1] Durrande et al. décomposent un noyau classique de processus gaussien (un Matern) en deux sous-noyaux : un noyau permettant de décrire une composante aléatoire périodique et une composante résiduelle. Cette décomposition s'inspire de la décomposition en série de Fourier et permet, via une comparaison des deux sous-noyaux, d'évaluer à quel point les données interpolées sont périodiques. Notre but est de présenter une extension de cette méthode en utilisant les ondelettes, nous permettant ainsi d'obtenir un noyau non stationnaire. Notre objectif final est de pouvoir à la fois extraire et modéliser des composantes non stationnaires dans des signaux bruités afin de les caractériser et les simuler