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Author
Date
2015Type
- Journal Article
ETH Bibliography
yes
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Abstract
This paper introduces a matrix analog of the Bessel processes, taking values in the closed set E of real square matrices with nonnegative determinant. They are related to the well-known Wishart processes in a simple way: the latter are obtained from the former via the map x↦x⊤x. The main focus is on existence and uniqueness via the theory of Dirichlet forms. This leads us to develop new results of potential theoretic nature concerning the space of real square matrices. Specifically, the function w(x)=|detx|α is a weight function in the Muckenhoupt Ap class for −1 <α≤0 (p=1) and −1<α<p−1 (p>1). The set of matrices of co-rank at least two has zero capacity with respect to the measure m(dx)=|detx|αdx if α>−1, and if α≥1 this even holds for the set of all singular matrices. As a consequence we obtain density results for Sobolev spaces over (the interior of) E with Neumann boundary conditions. The highly non-convex, non Lipschitz structure of the state space is dealt with using a combination of geometric and algebraic methods. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000102185Publication status
publishedExternal links
Journal / series
Electronic Journal of ProbabilityVolume
Pages / Article No.
Publisher
Institute of Mathematical StatisticsSubject
Matrix-valued process; Bessel process; Wishart process; Muckenhoupt weight; Positive determinant matrix; Reflecting boundary conditionOrganisational unit
09546 - Larsson, Martin (ehemalig) / Larsson, Martin (former)
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ETH Bibliography
yes
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