Hitting probability for Reflected Brownian Motion at Small Target

We derive the asymptotic behavior of hitting probability at small target of size $O(\epsilon)$ for reflected Brownian motion in domains with suitable smooth boundary conditions, where the boundary of domain contains both reflecting part, absorbing part and target. In this case the domain could be localized near the target and explicit computations are possible. The asymptotic behavior is only related to $\epsilon$ up to some multiplicative constants that depends on the domain and boundary conditions.