= Critical quadratic potential for the Ornstein-Uhlenbeck generator
{title2=$u(t,x)=\frac{e^{\lambda dt/2}}{(1+\lambda t)^{d/2}}\exp\left(\frac{\lambda^2t|x|^2}{2(1+\lambda t)}\right)$}
For $\mathcal L=\tfrac12\Delta-\lambda x\cdot\nabla$ and potential $V=\lambda^2|x|^2/2$, the <Feynman-Kac formula> with initial value one gives $u=\mathbb E_x^{OU}e^{\int_0^tV(X_s)ds}$. 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 $\lambda\ge0$; when $\lambda<0$ the path expectation diverges at and beyond $t=-1/\lambda$.
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