I'm wondering what happens to the following Borel integral when t goes to infinity:
$$\int_0^1 e^{B_t-yt}\mu(dy)$$
where $B_t$ is the standard Brownian motion and $\mu$ is a finite positive Borel measure and 0 is the left boundary of $\mu$.
I'm pretty sure the integral diverges a.s., i.e. its limsup goes to infinity a.s., and the liminf goes to 0 a.s. In fact I can show that's the case if the measure $\mu$ has a Dirac mass at 0. Unfortunately, I can't prove this in general. Any ideas?