Explicit formula for evolution semigroup for diffusion in Hilbert space

Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2018. Vol. 21. No. 4. P. 1850025-1–1850025-35.

Remizov I.

A parabolic partial differential equation u 0 t (t, x) = Lu(t, x) is considered, where L is a linear second-order differential operator with time-independent (but dependent on x) coefficients. We assume that the spatial coordinate x belongs to a finite- or infinitedimensional real separable Hilbert space H. 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 L. 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 H 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 H. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in L 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

Keywords: heat equation, Hilbert space, diffusion equation, Feynman formula