Brownian motion and the heat kernels of Iwasawa \(N A\)-type groups (Q2730494)

Let \(G\) be a semisimple Lie group with finite center, and let \(G = NAK\) be the Iwasawa decomposition of \(G\). Using a Riemannian metric constructed from the Killing form and a Cartan involution, one can formulate the heat equation \(\Delta u =\frac{\partial u}{\partial t}\) on \(NA\). Here \(\Delta\) is the Laplace-Beltrami operator associated to the Riemannian metric. The author proves a decomposition theorem for a stochastic differential equation on \(NA\) related to this heat equation. This theorem provides a means for calculating the heat kernel on \(NA\). The author applies these results to give a short calculation of the heat kernel of real hyperbolic \(n\)-space.