Representation of Green's function for the heat equation on a compact Lie group
type: Item
instance of: scholarly article
title: Representation of Green's function for the heat equation on a compact Lie group
zbMATH Open document ID: 0982.60069
DOI: 10.1007/BF02676670
author: M. G. Smirnova
published in: Mathematical Notes
publication date: 2000-11-16T00:00:00Z
review text: Let \(G\) be a compact connected Lie group of dimension \(m\). Then \(G\) may be realized as a subgroup of \(U(N)\) for some \(N\). For \(N\times N\) matrices \(A\) and \(B\), the formula \(\langle A,B\rangle= \text{trace}(AB^*)\) extends uniquely from \(\text{Lie}(G)\) to a bi-invariant Riemannian metric on \(G\). Let \(\Delta\) be the Laplacian (acting on functions) associated to this metric. The author considers the Cauchy initial value problem \[ {\partial f\over\partial t}={1\over 2} \Delta f,\qquad f|_{t=0}= f_0, \] with initial data \(f_0\in C^2(G)\). For \(N\times N\) matrices \(X\) and \(Y\), set \[ p(X, Y, h)= {C_h\over (\sqrt{2\pi h})^m} e^{-\|X-Y\|^2/2h}, \] where \(\|X\|^2= \langle X,X\rangle\) and the constant \(C_h\) is chosen so that \(\int_G p(X, E,h) dX= 1\); here \(E\) is the identity matrix and \(dX\) is Haar measure on \(G\). Note that \(p(X,E,h)\) is just an (\(h\) dependent) multiple of the standard Gaussian solution to the Euclidean heat equation on \(\mathbb{C}^{N^2}\) based at \(E\). The author proves the following representation for the solution \(f\): \[ f(t,X)= \lim_{\substack{ \max h_i\to 0\\ h_1+\cdots+ h_n= 1}} \int_G\cdots\int_G f_0(X)p(X_1, X_2, h_1)p(X_2, X_3, h_2)\cdots p(X_n, X, h_n) dX_1\cdots dX_n. \] Thus, \(f\) may be obtained as a limit of convolutions with the (scaled) Euclidean heat kernel \(p\).
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 31-XX
Mathematics Subject Classification ID: 58J99
zbMATH DE Number: 1531326
zbMATH Keywords: partial differential equation
zbMATH Keywords: heat equations
zbMATH Keywords: compact Riemannian manifold
zbMATH Keywords: Ricci curvature
zbMATH Keywords: compact Lie group
zbMATH Keywords: stochastic process
MaRDI profile type: Publication
Wikidata QID: Q115391893
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description: scientific article; zbMATH DE number 1531326