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

ofer zeitouni · Accepted Answer

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).

Large deviation for Brownian occupation time

One Answer

Add your own answers!

Ask a Question