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On the existence of fractional brownian fields indexed by manifolds

Abstract : The aim of the thesis is the study of the existence of fractional Brownian fields indexed by Riemannian manifolds. Those fields inherit key properties of the classical fractional Brownian motion (sample paths with self-similarity of adjustable parameter H, stationary increments), while allowing to consider applications with data indexed by a space which can be for example curved or with a hole. The existence of those fields is only insured when the quantity 2H is inferior or equal to the fractional index of the manifold, which is known only in a few cases. In a first part we give a necessary condition for the fractional Brownian field to exist. In the case of the Brownian field (corresponding to H=1/2) indexed by a manifold with minimal closed geodesics this condition happens to be very restrictive. We give several nonexistence results in this situation. In particular we show that there exists no Brownian field indexed by a nonsimply connected compact manifold. Our necessary condition also gives a short proof of an expected result: we prove the nondegeneracy of fractional Brownian fields indexed by the real hyperbolic spaces. In a second part we show that the fractional index of the cylinder is null, which gives a totally degenerate case. We deduce from this result that the fractional index of a metric space is noncontinuous with respect to the Gromov-Hausdorff convergence. We generalise this result about the cylinder to a Cartesian product with a closed minimal geodesic. Furthermore we give a bound of the fractional index of surfaces asymptotically close to the cylinder in the neighbourhood of a closed minimal geodesic.
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Nil Venet. On the existence of fractional brownian fields indexed by manifolds. Algebraic Geometry [math.AG]. Université Paul Sabatier - Toulouse III, 2016. English. ⟨NNT : 2016TOU30377⟩. ⟨tel-01825845⟩

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