Cross-Post from Math.SE
Let $(B_t)$ be a standard $d$-dimensional Brownian motion. It is well-known that $$\mathbb P(\sup_{s\in[0,t]}|B_s|\ge \alpha) \le 4de^{-\alpha^2/2dt}.$$ One possibility to obtain such a result is Schilder's theorem. This, a priori, only gives information on small time scales, but extends to arbitrary times by the scale invariance of a ball.
I am now interested in getting a similar probability tail estimate for arbitrary open connected domains $\Omega$, with Gaussian decay in the path distance (the infimum over all rectifiable paths in my domain) and Brownian motion stopped, when it leaves the domain. The large deviation principle given by Schilder's theorem in this case immediately implies an upper bound, but only for small times.
What is known for finite times, especially in terms of explicit upper bounds on the probability? Any comments and references are warmly welcome.
