  A parabolic partial differential equation $u'_t(t,x)=Lu(t,x)$ is considered,
where $L$ is a linear second-order differential operator with time-independent
coefficients, which may depend on $x$. We assume that the spatial coordinate
$x$ belongs to a finite- or infinite-dimensional real separable Hilbert space
$H$.
  Assuming the existence of a strongly continuous resolving semigroup for this
equation, we construct a representation of this semigroup by a Feynman formula,
i.e. we write it in the form of the limit of a multiple integral over $H$ as
the multiplicity of the integral tends to infinity. This representation gives a
unique solution to the Cauchy problem in the uniform closure of the set of
smooth cylindrical functions on $H$. Moreover, this solution depends
continuously on the initial condition. In the case where the coefficient of the
first-derivative term in $L$ vanishes we prove that the strongly continuous
resolving semigroup exists (this implies the existence of the unique solution
to the Cauchy problem in the class mentioned above) and that the solution to
the Cauchy problem depends continuously on the coefficients of the equation.
