Large deviation for Brownian occupation time

Question

I am asking for reference about the large deviation principle (LDP) for the occupation time of a Brownian motion/bridge. Let $f:\mathbb{R} \to \mathbb{R}$ be smooth and compactly supported. My question is: What is the LDP of $$\lambda^{-1} \int_0^\lambda f(B_s) ds, \quad\lambda \to \infty \ $$ Here, $B_s, s \in [0, \lambda]$ can be either a Brownian motion or a Brownian bridge with endpoints zero.

I find that Theorem 3.1 of http://www.brunoremillard.com/Papers/wiener.pdf provides a related result. However, I do not think this note is very well-written and I am asking for a better reference. Thank you!

Answer 1

This is precisely the Donsker-Varadhan LDP, coupled with an application of the contraction principle. Namely, the rate function is $$I(x)=\inf\{ J(\mu): \int f d\mu =x\}$$ where $J$ is the Donsker-Varadhan rate function. Look at the series of Donsker-Varadhan papers from 1975 (#I is the one you need) and any text on large deviations theory for the contraction principle (or the wikipedia page).