We consider Feynman-type approximations to functional integrals with respect to the Brownian sheet on a compact connected Lie group M, equipped with the associated bi-invariant Riemannian metric. The distribution of the Brownian sheet is represented as a weak limit of Gaussian-type measures on finite Cartesian products of the considered manifold.
For a two-parameter Brownian motion, that can also be described as a one-parameter process taking values in the functional space C([0,1],M), there is no established definition in the general case of an arbitrary compact manifold, however in the case when M is a compact connected Lie group equipped with a bi-invariant Riemannian metric there exists a process that has standard properties of a Brownian sheet on Rd. In this work we represent the distribution of the Brownian sheet with values in a compact Lie group as the weak limit of measures, that are constructed as the images of Gaussian-type measures on finite-dimensional spaces. To do that, the parametric set is divided on a finite number of rectangles of a small diameter, a continuous map from [0,1]2 to M is constructed as an interpolation based on its values in the nodes. Thus we get an inclusion Mn to C([0,1]2,M). Constructing the approximating measures on Mn we consider their images in C([0,1]2,M) under this inclusion and prove that their weak limit as diameters tend to zero coincides with the distribution of the Brownian sheet on M.