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Projections of regularized Brownian motions by the action of a Hilbertian Lie group. (Projections de mouvements Browniens régularisés via l’action d’un groupe de Lie Hilbertien.) (French) Zbl 0856.60013

Summary: We present here recent results established by the author and M. Arnaudon [Stochastics Stochastics Rep. 53, No. 1/2, 81-107 (1995; Zbl 0853.58111), “Regularisable and minimal orbits for group actions in infinite dimensions” (Manuscript, 1995) and “The geometric and physical relevance of some stochastic tools on Hilbert manifolds” (Manuscript, 1995)] – which were partly motivated by gauge theoretical problems – concerning projections of a class of martingales (which we shall refer to as regularized Brownian motions) locally defined by a stochastic differential equation on a Hilbert manifold and projected via the isometric action of an infinite-dimensional Lie group on this manifold. With the help of geometric concepts such as the notion of minimality of orbits, entended to the infinite-dimensional case, we generalize to the setting of infinite-dimensional manifolds, well-known results in the finite-dimensional case, such as the fact that a Brownian motion projects (via an isometric action) onto a Brownian motion when the orbits are minimal.

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60K40 Other physical applications of random processes

Citations:

Zbl 0853.58111
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References:

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[3] Arnaudon, M., Paycha, S., The geometric and physical relevence of some stochastic tools on Hilbert manifolds, Manuscript 1995
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