First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path

Volume growth and heat kernel estimates for the continuum random tree

“…The distribution of this random variable is known explicitly, see [10], and the above tail estimate is readily deduced from the expression that is given there. Using the fact that 1 −x ≤ e −x for x ≥ 0 and summing over n, we have…”

Packing measure, and its evaluation for a Brownian path

“…In [2] it was shown that </>(s) = i2log|logj| is the correct function to give the Hausdorff measure of a Brownian trajectory in Rd (d > 3). In §7 we show that \p(s) = 52/log|logi| is the correct function to make the ¡//-packing measure of the trajectory finite and positive.…”

Random Walks and A Sojourn Density Process of Brownian Motion

“…For r1 fixed, the distribution of this random quantity is completely given in [1]. Let us remark, finally, that in [1] it is shown that for JV > 2 the distribution of the total sojourn time in a sphere of radius rl is identical with that of the first passage time to the sphere of radius í-j in dimension N-2.lt is to be noted here that this first passage time may be represented for JV > 4 by letting r2 = r, and r0 = 0 in (4.8), which gives the expression (4.10) T 2((/V -4)(x + r^N~4\N -4)-1))-2-2/(N_4) X(x)dx.…”