Solution of the Cauchy problem for the heat equation on the Heisenberg group and the Wiener integral

The issues related to applications of functional integrals to evolution equations are studied. In particular, this is the problem of representation of solutions to the Cauchy problem for the heat equation in the three-parameter Heisenberg group $H_3(\mathbb{R})$ in terms of Wiener integral in the space of trajectories from $C[0,t]\times C[0,t]$.