Non-asymptotic estimates of invariant measures and regularisation by a degenerate noise for a chain of ordinary differential equations

Abstract : In the first part of this thesis, we aim to estimate the invariant distribution of an ergodic process driven by a Stochastic Differential Equation. The ergodic theorem suggests us to consider the empirical measure associated with a discretization scheme of the process which can be regarded as a discretization of the occupation measure of the process.Lamberton and Pagès introduced an algorithm of discretization with decreasing time steps which allows the convergence of the empirical measure toward the invariant distribution of the process, they also provide a central limit theorem (CLT) which asymptotically quantifies the deviations between these both measures.We establish non-asymptotic concentration inequality for the empirical measure deviations (in accordance with the previously mentioned CLT), and also we give some controls of the solution of the associated Poisson equation which is useful for this concentration inequalities.In a second part, we establish some Schauder controls associated with parabolic equations related with a degenerate stochastic system, where the drift is a vector field satisfying a weak Hörmander condition like.But we aim to suppose only the minimal H"older regularity.This work is an extension of the estimates given by Delarue and Menozzi (2010). Finally, our approach allows us to proof the strong uniqueness of the considered stochastic equation in a H"older regularity framework. Our results extend the controls of Chaudru de Raynal (2017) for the dimension equal to 2.
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Contributor : Igor Honoré <>
Submitted on : Thursday, December 20, 2018 - 10:55:26 AM
Last modification on : Saturday, January 12, 2019 - 1:13:39 AM

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  • HAL Id : tel-01961770, version 1

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Igor Honoré. Non-asymptotic estimates of invariant measures and regularisation by a degenerate noise for a chain of ordinary differential equations. Analysis of PDEs [math.AP]. Université Paris-Saclay, 2018. English. ⟨tel-01961770⟩

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