My question is how to make sense of the fact that the Langevin noise correlation $\langle \eta(t) \eta(t') \rangle = \Gamma \delta(t-t')$ diverges when $t'=t$. In particular, consider the Langevin equation
$\begin{equation} \gamma \dot{x} = - \frac{dV(x)}{dt} + \eta(t). \end{equation} $
Suppose now that I want to compute $\langle \dot{x}(t)^2 \rangle$. This gives
$\begin{equation} \langle \dot{x}(t)^2 \rangle = - \frac{1}{\gamma^2} \langle (\frac{dV(x(t))}{dt})^2 \rangle + \frac{1}{\gamma^2} \langle \eta(t) \eta(t) \rangle, \end{equation} $
However, naively the last term would be divergent, whereas intuitively $\langle \dot{x}^2 \rangle = \langle v^2 \rangle$ should be finite.