Dissertation/ Thesis
Probabilistic and statistical analysis of diffusion systems in presence of noise ; Analyse statistique et probabiliste de systèmes diffusifs en présence de bruit
| العنوان: | Probabilistic and statistical analysis of diffusion systems in presence of noise ; Analyse statistique et probabiliste de systèmes diffusifs en présence de bruit |
|---|---|
| المؤلفون: | Maillet, Raphaël |
| المساهمون: | CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris Sciences et Lettres (PSL)-Université Paris Sciences et Lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Université Paris sciences et lettres, Marc Hoffmann, Pierre Cardaliaguet |
| المصدر: | https://theses.hal.science/tel-04828513 ; Analysis of PDEs [math.AP]. Université Paris sciences et lettres, 2024. English. ⟨NNT : 2024UPSLD025⟩. |
| بيانات النشر: | CCSD |
| سنة النشر: | 2024 |
| المجموعة: | Université Paris-Dauphine: HAL |
| مصطلحات موضوعية: | Ergodic Diffusion Processe, Invariant Measure, Noise Reduction, Stochastic Fokker-Planck, Kernel Density Estimation, Non-Parametric Estimation, Estimation de densité par noyau, Estimation non-paramétrique, Mesure invariante, Équations de Fokker-Planck stochastiques, Processus de diffusion ergodique, Bruit commun, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] |
| الوصف: | This thesis deals with the long-time behavior of stochastic Fokker-Planck equations with additive common noise and presents statistical methods for estimating the invariant measure of multidimensional ergodic diffusion processes from noisy data. In the first part, we analyze stochastic Fokker-Planck Partial Differential Equations (SPDEs), obtained as the mean-field limit of interacting particle systems influenced by both idiosyncratic and common Brownian noises. We establish conditions under which the addition of common noise restores uniqueness if the invariant measure. The main challenge arises from the finite-dimensional nature of the common noise, while the state variable — interpreted as the conditional marginal law of the system given the common noise — operates within an infinite-dimensional space. We demonstrate that uniqueness is restored if the mean field interaction term attracts the system towards its conditional mean given the common noise, particularly when the intensity of the idiosyncratic noise is small. In the second part, we develop a new statistical methodology using kernel density estimation to effectively approximate the invariant measure from noisy observations, highlighting the crucial role of the underlying Markov structure in the denoising process. This method involves a pre-averaging technique that proficiently reduces the intensity of the noise while maintaining the analytical characteristics and asymptotic properties of the underlying signal. We investigate the convergence rate of our estimator, which depends on the anisotropic regularity of the density and the intensity of the noise. We establish noise intensity conditions that allow for convergence rates comparable to those in noise-free environments. Additionally, we demonstrate a Bernstein concentration inequality for our estimator, leading to an adaptive procedure for selecting the kernel bandwidth. ; Cette thèse traite du comportement en temps long des équations stochastiques de Fokker-Planck en présence d’un bruit commun ... |
| نوع الوثيقة: | doctoral or postdoctoral thesis |
| اللغة: | English |
| Relation: | NNT: 2024UPSLD025 |
| الاتاحة: | https://theses.hal.science/tel-04828513 https://theses.hal.science/tel-04828513v1/document https://theses.hal.science/tel-04828513v1/file/2024UPSLD025.pdf |
| Rights: | info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.FC5B979D |
| قاعدة البيانات: | BASE |
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