Let be a Brownian motion valued in the complex projective space . Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of and of , and express them through Jacobi polynomials in the simplices of and respectively. More generally, the distribution of may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group yet computations become tedious. We also revisit the approach initiated in [13] and based on a partial differential equation (hereafter pde) satisfied by the Laplace transform of the density. When , we invert the Laplace transform and retrieve the expression already derived using spherical harmonics. For general , integrations by parts performed on the pde lead to a heat equation in the simplex of .
@article{AMBP_2014__21_2_1_0, author = {Demni, Nizar}, title = {Distributions of truncations of the heat kernel on the complex projective space}, journal = {Annales math\'ematiques Blaise Pascal}, pages = {1--20}, publisher = {Annales math\'ematiques Blaise Pascal}, volume = {21}, number = {2}, year = {2014}, doi = {10.5802/ambp.339}, mrnumber = {3322612}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ambp.339/} }
TY - JOUR AU - Demni, Nizar TI - Distributions of truncations of the heat kernel on the complex projective space JO - Annales mathématiques Blaise Pascal PY - 2014 SP - 1 EP - 20 VL - 21 IS - 2 PB - Annales mathématiques Blaise Pascal UR - http://www.numdam.org/articles/10.5802/ambp.339/ DO - 10.5802/ambp.339 LA - en ID - AMBP_2014__21_2_1_0 ER -
%0 Journal Article %A Demni, Nizar %T Distributions of truncations of the heat kernel on the complex projective space %J Annales mathématiques Blaise Pascal %D 2014 %P 1-20 %V 21 %N 2 %I Annales mathématiques Blaise Pascal %U http://www.numdam.org/articles/10.5802/ambp.339/ %R 10.5802/ambp.339 %G en %F AMBP_2014__21_2_1_0
Demni, Nizar. Distributions of truncations of the heat kernel on the complex projective space. Annales mathématiques Blaise Pascal, Volume 21 (2014) no. 2, pp. 1-20. doi : 10.5802/ambp.339. http://www.numdam.org/articles/10.5802/ambp.339/
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