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We consider a singularly perturbed boundary value problem( −  ε 2  ∆ + ∇ V · ∇ )   u ε  = 0in  Ω ,  u ε  = f  on  ∂ Ω , f ∈  C ∞   ( ∂ Ω )  .The function V is supposed to be sufficiently smooth and to have the only minimum in the domain    Ω   . This minimum can degenerate. The potential V has no other stationary points in    Ω    and its normal derivative at the boundary is non-zero. Such a problem arises in studying Brownian motion governed by overdamped Langevin dynamics in the presence of a single attracting point. It describes the distribution of the points at the boundary∂ Ω    , at which the trajectories of the Brownian particle hit the boundary for the first time. Our main result is a complete asymptotic expansion foru εasε → + 0    . This asymptotic is a sum of a termK ε   Ψ εand a boundary layer, whereΨ εis the eigenfunction associated with the lowest eigenvalue of the considered problem andK εis some constant. We provide complete asymptotic expansions for bothK εandΨ ε    ; the boundary layer is also an infinite asymptotic series power in    ε   . The error term in the asymptotics foru εis estimated in various norms.

Complete Asymptotics for Solution of Singularly Perturbed Dynamical Systems with Single Well Potential