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Consider the standard one-dimensional Brownian motion $B_t$. Set $$X_t = \int_0^t\mathbf{1}_{[0, 1]}(B_s)ds.$$ Here $\mathbf{1}_{[0, 1]}(\cdot)$ is the indicator function of the interval $[0, 1]$. $X_t$ denotes the total time spent by Brownian motion in $[0, 1]$ before time $t$, which can also be written using the Brownian local time as $$X_t = \int_0^1L^a(t)da.$$

Question:

Can we compute the distribution of $X_t$? If we can't get the specific expression of its density, is there any method to get the decay estimate (lower and upper bound) of its density?

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1 Answer 1

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I know this is late response, but you might be interested in the theory of "Sojourn Times", in particular the sojourn times of a Brownian motion.

You might then be interested in the moments of the brownian local times, and the generalization to intervals. What you are interested in is the time spent in an interval by a Brownian motion, which can also be obtained by the time spent in half the interval by a reflecting Brownian motion. The results you seek can be found in the paper: "SOJOURN TIMES FOR THE BROWNIAN MOTION" by LAJOS TAKACS.

Hope this help.

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