Diffusion processes on solvable groups of upper triangular 2×2 matrices and their approximation

Diffusion processes on solvable groups of upper triangular 2 × 2 matrices and the approximation of diffusions on similar Lie groups by random walks with small step size are studied. Brownian motions on these groups are constructed by using the multiplicative stochastic integral. For a Riemannian volume on a Lie group, spectral indices are of the same nature as in the case of Abelian or nilpotent groups, although the geometric properties of these groups are substantially different. The fundamental difference between the return probabilities in the continuous and discrete models is caused by that the discrete subgroup is dense and very chaotically distributed in the group.