Let $X_t = e^{-\lambda t} \left(X_0 + \int _0^t e^{\lambda u} dW_u\right)$ where $(W_u)_{u \geqslant 0}$ is a Wiener process, $X_0$ random variable of law $\nu$ and independent of $\int _0^t e^{\lambda u} dW_u$.
I want to show that there is no meaningful distribution of the limit if $\lambda<0$. Using characteristic function and the independence I can show the limiting characteristic function is $0$ always, so it isn't a characteristic function. But does that tell me there can be no such measure?
I know the first term goes to infinity and the variance of the normal distribution of the second term as well, but I would prefer a better argument than that both those measures are respectively not finite or defined.