Stochastic invertibility and related topics

Abstract : I this work we investigate a notion of stochastic invertibility on Wiener space. Rougghly speaking a morphism of probability spaces with values on the Wiener space, which is further adapted, can be canonically associated to the laws of the solutions to some stochastic differential equations. One of the main properties of this morphism is to be invertible (i.e. to be an isomorphism of probability spaces) if and only if the underlying stochastic differential equation has a unique strong solution. Since it may be seen as a Brownian motion, we cal it the Brownian transform of the associated law, and we will study the invertibility of this Brownan transform. We will see that this notion, whose origins may be found in earlier results related to stochastic mechanics, extends and enlightens the notion of invertibility of adapted shifts on Wiener space which was investigated by Üstünel and Zakai in their recent papers, where this notion already appears clearly between the lines. Moreover, from the origin many problems arising in various fields are deeply related to this notion. This opens to a wide spectrum of applications, some of them being very concrete. We will investigate problems of various origins such as statistical physics, information theory, filtering, but also stochastic control and optimal transport. For instance, we will prove a very general result of pathwise uniqueness for the stochastic picture of euclidean quantum mechanics, and we will extend Shannon’s inequality to any abstract Wiener spaces. We also show how this notion of invertivility fits naturally in stochastic differencial geometry.
Keywords : Invertibility
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Theses
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Rémi Lassalle. Stochastic invertibility and related topics. General Mathematics [math.GM]. Télécom ParisTech, 2012. English. ⟨NNT : 2012ENST0030⟩. ⟨tel-01136574⟩

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