Asymptotic of heat kernels on projective spaces of large dimensions and on disk hypergroups. (English) Zbl 0922.60017
Let \(h^x_{t,n}=h_n(t,x,\cdot)\), where \(h_n\) is the heat kernel on the complex projective space \(\mathbb P_n(\mathbb C)\), \(x\in \mathbb P_n(\mathbb C)\), \(t>0\). Denote \(t(n)=(4n)^{-1}[\log (n-1)+c],\;c\in \mathbb R\). For each \(x\in \mathbb P_n(\mathbb C)\) the author finds the asymptotics of \(\| h^x_{t(n),n}-1\| _{1,U_n}\) for \(n\to \infty\). Here \(\| \cdot \| _{1,U_n}\) is the \(L_1\)-norm with respect to the uniform distribution on \(\mathbb P_n(\mathbb C)\). Similar results are obtained for the quaternionic projective space, symmetric spaces \(U(n)/U(n-1)\), and disk hypergroups. The proofs are based on the comparison of heat kernels with Poisson kernels, and the central limit theorem for Poisson kernels.
Reviewer: Anatoly N.Kochubei (Kiev)
MSC:
| 60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |
| 60F05 | Central limit and other weak theorems |
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 43A62 | Harmonic analysis on hypergroups |
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