Academic literature on the topic 'Ergodic Diffusion Processe'

Cette thèse traite du comportement en temps long des équations stochastiques de Fokker-Planck en présence d’un bruit commun additif et présente des méthodes statistiques pour estimer la mesure invariante des processus de diffusion ergodiques multidimensionnels à partir de données bruitées. Dans la première partie, nous analysons les équations différentielles partielles stochastiques de type Fokker-Planck non linéaires, obtenues comme la limite du champ moyen de systèmes de particules en interaction dirigés par des bruits browniens idiosyncrasiques et en présence de bruit commun. Nous établissons des conditions sous lesquelles l’ajout d’un bruit commun promet de restaurer l’unicité de la mesure invariante. La principale difficulté provient de la dimension finie du bruit commun, alors que la variable d’état- interprétée comme la loi marginale conditionnelle du système compte tenu du bruit commun - opère dans un espace de dimension infinie. Nous démontrons que l’unicité est rétablie dès lors que le terme d’interaction du champ moyen attire le système vers sa moyenne conditionnelle (par rapport au bruit commun), en particulier lorsque l’intensité du bruit idiosyncrasique est faible, qui sont des cas typiques de perte d’unicité en l’absence de bruit commun. Dans la deuxième partie, nous développons une méthodologie statistique afin d’approximer la mesure invariante d’un processus de diffusion à partir d’observations bruitées et discrètes de ce même processus. Cette méthode implique une technique de pré-moyennage des données qui réduit l’intensité du bruit tout en conservant les caractéristiques analytiques et les propriétés asymptotiques du signal sous-jacent. Nous étudions le taux de convergence de cet estimateur, qui dépend de la régularité anisotrope de la densité et de l’intensité du bruit. Nous établissons ensuite des conditions sur l’intensité du bruit qui permettent d’obtenir des taux de convergence comparables à ceux des cas sans bruit. Enfin, nous démontrons une inégalité de concentration de type Bernstein pour notre estimateur, ce qui promet de mettre en place une procédure adaptative pour la sélection de la fenêtre du noyau<br>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

This thesis is concerned with methodologies for the accurate quantitative modelling of molecular biological systems. The first part is devoted to the chemical Langevin equation (CLE), a stochastic differential equation driven by a multidimensional Wiener process. The CLE is an approximation to the standard discrete Markov jump process model of chemical reaction kinetics. It is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. We observe that the CLE is not a single equation, but a family of equations with shared finite-dimensional distributions. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m_1 pairs of reversible reactions and m_2 irreversible reactions, there is another, simple formulation of the CLE with only m_1+m_2 Wiener processes, whereas the standard approach uses 2m_1+m_2. Considerable computational savings are achieved with this latter formulation. A flaw of the CLE model is identified: trajectories may leave the nonnegative orthant with positive probability. The second part addresses the challenge when alternative, structurally different ordinary differential equation models of similar complexity fit the available experimental data equally well. We review optimal experiment design methods for choosing the initial state and structural changes on the biological system to maximally discriminate between the outputs of rival models in terms of L_2-distance. We determine the optimal stimulus (input) profile for externally excitable systems. The numerical implementation relies on sum of squares decompositions and is demonstrated on two rival models of signal processing in starving Dictyostelium amoebae. Such experiments accelerate the perfection of our understanding of biochemical mechanisms.

Controlled linear stochastic evolution equations driven by Lévy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Itô formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. The control problem is solved in the mean-value sense and, under selective conditions, in the pathwise sense. As examples, various parabolic type controlled SPDEs are studied. 1