From Feynman-Kac formula to Feynman integrals via analytic continuation

AbstractBy using a calculus based on Brownian bridge measures, it is shown that under mild assumptions on V (e.g. V is in the Kato class) the fundamental solution (FS) q (t,x,y) for the heat equation ∂tu = 12Δ − V)u can be represented by the Feynman-Kac formula. Furthermore, it has an analytic continuation in t over C+, where C+={z ∈ C, Re z > 0}, and q(ε + it,x,y) can be expressed via Wiener path integrals. For small ε > 0 it can be considered as an approximation of the FS for the Schrödinger equation ∂tψ=i(12Δ − V)ψ. We also give an estimate of q(t,x,y) for t ∈ C+