We present a general method of solving the Cauchy problem for
multidimensional parabolic (diffusion type) equation with variable coefficients
which depend on spatial variable but do not change over time. We assume the
existence of the $C_0$-semigroup (this is a standard assumption in the
evolution equations theory, which guarantees the existence of the solution) and
then find the representation (based on the family of translation operators) of
the solution in terms of coefficients of the equation and initial condition. It
is proved that if the coefficients of the equation are bounded, infinitely
smooth and satisfy some other conditions then there exists a solution-giving
$C_0$-semigroup of contraction operators. We also represent the solution as a
Feynman formula (i.e. as a limit of a multiple integral with multiplicity
tending to infinity) with generalized functions appearing in the integral
kernel.