Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 30 / 6 / a / Solution

The differential operator without its potential is the diffusion generator of the d-dimensional Ornstein-Uhlenbeck processIts explicit solution $X_s=e^{-\lambda s}x+{\int }_0^s{e}^{-\lambda (s-r)}d{B}_r$ defines its path law ${\mathbb{P}}^x$ on $C([0,t];{\mathbb{R}}^d)$. For the potential $V(x)={\lambda }^2|x{|}^2/2$ and initial function one, the Feynman-Kac formula gives the nonnegative path integralThe sign in the exponential is positive, because the PDE contains $+Vu$. One can first stop the diffusion in bounded balls, where the usual localized Feynman-Kac argument applies, and then examine finiteness of the resulting expectation. The critical quadratic potential for the Ornstein-Uhlenbeck generator below supplies that finiteness check.