Solution to a parabolic differential equation in Hilbert space via Feynman formula - parts I and II

TL;DR

This paper constructs a Feynman formula representation for solutions of parabolic PDEs in Hilbert spaces, providing a new way to solve and analyze such equations with potential applications in infinite-dimensional analysis.

Contribution

It introduces a Feynman formula approach to represent the semigroup solving parabolic PDEs in Hilbert spaces, establishing existence, uniqueness, and continuous dependence of solutions.

Findings

Constructed a Feynman formula representation for the semigroup.

Proved existence and uniqueness of solutions in the specified function space.

Showed continuous dependence of solutions on initial conditions and coefficients.

Abstract

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.