Wiener Measure For Nilpotent Matrix Groups

N. Wiener constructed a mathematical model of Brownian motion in which the basicprobabilities were the values of a measure defned on subsets of a space of continuous func-tions, then constructed Wiener measure and Wiener integral. This model made a perfectdescription for Brownian motion. Under the influence of Wiener’s work, R. Feynman andKac were devoted to the solution of Schr¨odinger equation and gave the expression of theexplicit solution.In this article, we established Wiener measure and Wiener integral for the nilpotentmatrix groups. First, we collected some fundamental results of the matrix groups andhomogeneous groups which used them to the nilpotent matrix groups. Based on theseresults, we established Wiener measure and Wiener integral. Second, as an application,we used it to obtain the explicit solution of Schr¨odinger equation on nilpotent matrixgroups.