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Étude d'une classe d'équations aux dérivées partielles stochastiques : Existence, unicité, comportement asymptotique

Abstract : In this thesis we investigate a class of stochastic, parabolic, semilinear partial differential equations driven by finite-dimensional Wiener processes. Such equations arise in problems from population dynamics and population genetics. In the first chapter we briefly analyze various models that have been proposed over the years in this field of research ; these models motivate the class of problems we study in this work. In the second chapter we develop a construction of Itô's stochastic integral for a class of Hilbert space-valued stochastic prpcesses. We also introduce a crass of auxiliary problems for which we prove the existence ànd the uniqueness of a variational solution. In the third chapter we prove a-comparison principle for the class of auxiliary problems we introduced in the second chapter, which allows us to prove the existence and the uniqueness of a variational solution to the original class of problems. Our method of proof of the comparison principle is set up in such a way that it allows us to unify various methods which have been devised over the years to prove comparison principles for stochastic, parabolic partial differential equations, Itô's stochastic differential equations as weIl as deterministic, parabolic, partial differential equations. In the fourth chapter we analyze the fine asymptotic behavior of the solutions to our class of problems when the time variable goes to infinity. In particular, we prove the existence of a global attractor and we unveil the mechanism where by the solutions approach the global attracter. We also show how to deterrnine explicitly the corresponding Lyapunov exponents when the nonlinearities of the noise terms are subordinated to the nonlinearities of the drift terms. Our analysis also allows us to exhibit a phenomenon of exchange of stability between the components of the global attractor. As an application of our results, we give two examples in the context of population genetics.
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Submitted on : Thursday, March 29, 2018 - 10:37:49 AM
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Benjamin Bergé. Étude d'une classe d'équations aux dérivées partielles stochastiques : Existence, unicité, comportement asymptotique. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2001. Français. ⟨NNT : 2001NAN10016⟩. ⟨tel-01746437⟩



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