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Diffusive processes on the Wasserstein space : coalescing models, regularization properties and McKean-Vlasov equations

Abstract : The aim of this thesis is to study a class of diffusive stochastic processes with values in the space of probability measures on the real line, called Wasserstein space if it is endowed with the Wasserstein metric W2. The following issues are mainly addressed in this work: how can we effectively construct a stochastic process satisfying diffusive properties with values in a space of infinite dimension? is there a form of uniqueness, in a strong or a weak sense, satisfied by some of those processes? do those diffusions own smoothing properties, e.g. regularization by noise of McKean-Vlasov equations or e.g. BismutElworthy integration by parts formulae? Chapter I introduces an alternative construction, by smooth approximations, of the particle system defined by Konarovskyi and von Renesse, hereinafter designed by coalescing model. The coalescing model is a random process with values in the Wasserstein space, following an Itô-like formula on that space and whose short-time deviations are governed by the Wasserstein metric, by analogy with the short-time deviations of the standard Brownian motion governed by the Euclidean metric. The regular approximation constructed in this thesis shares those diffusive properties and is obtained by smoothing the coefficients of the stochastic differential equation satisfied by the coalescing model. The main benefit of this variant is that it satisfies uniqueness results which are still open for the coalescing model. Moreover, up to small modifications of its structure, that smooth diffusion owns regularizing properties: this is precisely the object of study of chapters II to IV. In chapter II, an ill-posed McKean-Vlasov equation is perturbed by one of those smooth versions of the coalescing model, in order to restore uniqueness. A connection is made with recent results (Jourdain, Mishura-Veretennikov, Chaudru de Raynal-Frikha, Lacker, Röckner-Zhang) where uniqueness of a solution is proved when the noise is finite dimensional and the drift coefficient is Lipschitz-continuous in total variation distance in its measure argument. In our case, the diffusion on the Wasserstein space allows to mollify the velocity field in its measure argument and so to handle with drift functions having low regularity in both space and measure variables. Lastly, chapters III and IV are dedicated to the study, for a diffusion defined on the Wasserstein space of the circle, of the smoothing properties of the associated semi-group. Applying in chapter III the differential calculus on the Wasserstein space introduced by Lions, a Bismut-Elworthy inequality is obtained, controlling the gradient of the semi-group at those points of the space of probability measures that have a sufficiently smooth density. In chapter IV, a better explosion rate when time tends to zero is established under additional regularity conditions. This leads to a priori estimates for a PDE defined on the Wasserstein space and governed by the diffusion on the torus mentioned above, in the homogeneous case (chapter III) and in the case of a non-trivial source term (chapter IV).
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Victor Marx. Diffusive processes on the Wasserstein space : coalescing models, regularization properties and McKean-Vlasov equations. Probability [math.PR]. COMUE Université Côte d'Azur (2015 - 2019), 2019. English. ⟨NNT : 2019AZUR4065⟩. ⟨tel-02342939v2⟩

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