If we know the endpoints of the Brownian path, is there any theorem telling us if it can be contained within a ball a.s. (with probability one)? For example contained in two big enough balls (call it B) centered at the endpoints.
How about for Brownian paths (i.e. knowing only one endpoint)?
If not a.s., at least with a probability bounded by $\varepsilon(r)$, where r is the diameter of B covering parts of the path. So that as r increases the probability decrease.
related discussion can be found there: Confidence band for Brownian Motion with uniformly distributed hitting position
I think this is an interesting problem because we can then talk about the probability of hitting a set A (trapping problem).
Here is even more discussion: http://wrap.warwick.ac.uk/32145/1/WRAP_Kendall_KendallMarinRobert-2007.pdf
It turns out this is an open question. But feel free to post any additional info like references.
Thanks
Since the distribution of the values at the midpoint of the bridge (as anywhere else) is a normal distribution, the probability for arbitrarily large values is small but positive.
As is well known, $B_t\sim N(0,t)$ and the bridge has values $B_t-\frac{t}{T}B_T$ with mean value $0$ and variance $$ \Bbb E(B_t-\tfrac{t}{T}B_T)^2=\Bbb E((1-\tfrac{t}{T})B_t-\tfrac{t}{T}(B_T-B_t))^2=t(1-\tfrac{t}{T}) $$ To quantify the probability for large values, see quantiles of the normal distribution and their asymptotic approximations.