I am just working on a paper by Shinzo Watanabe on "Asymptotic Evaluations of Wiener Functional expactations":
$μ(dx)$ is the $d$-dimensional Gaussian distributions
So that is the interesting asymptotic relation which goes back to the Laplace method. I unterstood it in one Dimension and I think also in the multiveriat case without that addtional function $g(x)$. For the proof, Watanabe gives the following structure:
I also found out how to deal with $I_1$ but I simply don't know how he Comes to that estimate about $I_2$. How can one us the fact, that the amount of $g(x)=O(\exp(Kx^2))$ if $K>0$. I thought about proofing it by showing that the Gaussian measure fullfills a Large Deviation Principle and then use Varadahans Lemma. But I can't do it with the additional $g(x)$. Does any of you know a way to Show that estimate or has a good reference for the multivariat Laplace method?