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Equations d'évolution stochastiques dans les espaces de Banach : unicités abstraites, propriété de Markov forte, équations hyperboliques

Abstract : This work consists of four chapters on some aspects of stochastic semilinear evolution equations (SPDE) in Banach spaces. The first chapter deals with different notions of uniqueness and existence (such as pathwise uniqueness, uniqueness in law, strong and weak existence) and the relations between them. We present an alternative construction of the stochastic integral in Banach spaces and we prove Burkholder's inequality, Fubini's theorem, the Chojnowska-Michalik theorem and Girsanov's theorem. We prove distribution preserving theorems for Bochner integrals, stochastic integrals and measurable selectors as well. The second chapter regards the Brownian representations of local cylindrical martingales in Banach spaces and the martingale problem in infinite dimensions. We use these results for illustrating the role of the notion "well-posedness" and for showing that weak existence and uniqueness in law for the equation in question imply the strong Markov property of the solutions. The third and the fourth chapter treats second order hyperbolic SPDE's driven by a spatially homogeneous Wiener process. We present sufficient conditions on the coefficients for the equation to have global strong and weak solutions, and we prove that the solutions propagate at finite speed.
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Martin Ondrejat. Equations d'évolution stochastiques dans les espaces de Banach : unicités abstraites, propriété de Markov forte, équations hyperboliques. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2003. Français. ⟨NNT : 2003NAN10046⟩. ⟨tel-01748131⟩

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