  This short communication (preprint) is devoted to mathematical study of
evolution equations that are important for mathematical physics and quantum
theory; we present new explicit formulas for solutions of these equations and
discuss their properties. The results are given without proofs but the proofs
will appear in the longer text which is now under preparation.
  In this paper, infinite-dimensional generalizations of the Euclidean analogue
of the Schr\"odinger equation for anharmonic oscillator are considered in the
class of measures. The Cauchy problem for these equations is solved. In
particular cases, explicit formulas for fundamental solutions are obtained,
which are a generalization of the Mehler formula, and the uniqueness of the
solution with certain properties is proved. An analogue of the
Ornstein-Uhlenbeck measure is constructed. The definition of Gaussian
semigroups is given and their connection with the considered parabolic
equations is described.
