# Large deviation for Brownian occupation time

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
$$lambda^{-1} int_0^lambda f(B_s) ds, quadlambda to infty$$
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!

MathOverflow Asked by lye012 on January 26, 2021

This is precisely the Donsker-Varadhan LDP, coupled with an application of the contraction principle. Namely, the rate function is $$I(x)=inf{ J(mu): int f dmu =x}$$ 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).

Correct answer by ofer zeitouni on January 26, 2021

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