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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?

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    The integral does, in general, not diverge; for instance if $\mu(dy)$ is a Dirac measure at $y=1/2$, then the integral converges to $0$ as $t \to \infty$. – 2017-02-08
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    ...And probably for every $\mu$ such that $\mu(\{0\})=0$. – 2017-02-08
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    Thanks for the feedback. I added the condition that 0 is the left boundary of the measure $\mu$. It's not clear to me how to show convergence when $\mu(\{0\})=0$ and the left boundary of $\mu$ is 0. Thanks! – 2017-02-08
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    with the above comments saz and did hint to the law of the iterated logarithm for Brownian motion look [here](https://en.wikipedia.org/wiki/Wiener_process#Law_of_the_iterated_logarithm). In any case where $y\neq0$ $yt$ grows faster than brownian motion...(also you might want to use '@' if you adress somebody in the comments then they will be able to notice your response). – 2017-02-09

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