@prefix dcat: <http://www.w3.org/ns/dcat#> .
@prefix dct: <http://purl.org/dc/terms/> .
@prefix foaf: <http://xmlns.com/foaf/0.1/> .
@prefix vcard: <http://www.w3.org/2006/vcard/ns#> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

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    dct:description """
              Let $(U_t)_{t \\geq 0}$ be a Brownian motion valued in the complex projective space $\\mathbb{C}P^{N-1}$. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $|U_t^{1}|^2$ and of $(|U_t^{1}|^2, |U_t^2|^2)$, and express them through Jacobi polynomials in the simplices of $\\mathbb{R}$ and $\\mathbb{R}^2$ respectively. More generally, the distribution of $(|U_t^{1}|^2, \\dots, |U_t^k|^2), 2 \\leq k \\leq N-1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $\\mathcal{U}(N-k+1)$ yet computations become tedious. We also revisit the approach initiated in \\cite{Nec-Pel} and based on a partial differential equation (hereafter pde) satisfied by the Laplace transform of the density. When $k=1$, we invert the Laplace transform and retrieve the expression derived using spherical harmonics. For general $1 \\leq k \\leq N-2$, the integrations by parts performed on the pde lead to a heat equation in the simplex of $\\mathbb{R}^k$.
            """ ;
    dct:identifier "hal-00979479" ;
    dct:issued "2026-05-05T14:31:54.465576"^^xsd:dateTime ;
    dct:language "en" ;
    dct:modified "2026-05-05T14:31:54.465581"^^xsd:dateTime ;
    dct:publisher <https://rec.harvest-normandie.data4citizen.com/organization/cce9db95-46d9-4dc2-84b6-764215d0a002> ;
    dct:title "A stationary process associated with the Dirichlet distribution arising from the complex projective space" ;
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    dcat:keyword "infoeu-reposemanticspreprint",
        "mathmath-prmathematics-mathprobability-mathpr",
        "preprints-working-papers-" ;
    dcat:landingPage <https://hal.science/hal-00979479> .

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    dct:issued "2026-05-05T14:31:54.468335"^^xsd:dateTime ;
    dct:modified "2026-05-05T14:31:54.460084"^^xsd:dateTime ;
    dct:title "A stationary process associated with the Dirichlet distribution arising from the complex projective space" ;
    dcat:accessURL <https://hal.science/hal-00979479> .

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