Representation of Green's function for the heat equation on a compact Lie group
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 \(p\).
scientific article; zbMATH DE number 1531326