A parabolic partial differential equation [Formula: see text] is considered, where [Formula: see text] is a linear second-order differential operator with time-independent (but dependent on [Formula: see text]) coefficients. We assume that the spatial coordinate [Formula: see text] belongs to a finite- or infinite-dimensional real separable Hilbert space [Formula: see text]. The aim of the paper is to prove a formula that expresses the solution of the Cauchy problem for this equation in terms of initial condition and coefficients of the operator [Formula: see text]. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup using a Feynman formula (i.e. we write it in the form of a limit of a multiple integral over [Formula: see text] as the multiplicity of the integral tends to infinity), which gives us a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on [Formula: see text]. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in [Formula: see text] is zero, we prove that the strongly continuous resolving semigroup indeed exists (which implies the existence of a 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.
The behavior of solutions of non-linear differential equations in Hilbert space. I