Critical quadratic potential for the Ornstein-Uhlenbeck generator ($u(t,x)=\frac{e^{\lambda dt/2}}{(1+\lambda t{)}^{d/2}}\exp \left(\frac{{\lambda }^{2}t|x{|}^2}{2(1+\lambda t)}\right)$)

For $\mathcal{L}=\frac{1}{2}\Delta -\lambda x\cdot \nabla$ and potential $V={\lambda }^2|x{|}^2/2$, the Feynman-Kac formula with initial value one gives $u={\mathbb{E}}_x^{OU}{e}^{{\int }_0^{t}V(X_s)ds}$. The Ornstein-Uhlenbeck likelihood relative to Wiener measure cancels the quadratic time integral, leaving a Gaussian endpoint integral. Completing the square gives the displayed answer for $1+\lambda t>0$. It is finite for all nonnegative times when $\lambda \ge 0$; when $\lambda <0$ the path expectation diverges at and beyond $t=-1/\lambda$.