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

Denis I. Borisov and Oskar A. Sultanov
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Denis I. Borisov: Department of Differential Equations, Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Chernyshevsky str. 112, 450008 Ufa, Russia
Oskar A. Sultanov: Department of Differential Equations, Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Chernyshevsky str. 112, 450008 Ufa, Russia

Mathematics, 2020, vol. 8, issue 6, 1-17

Abstract: We consider a singularly perturbed boundary value problem ( − ε 2 ? + ∇ V · ∇ ) u ε = 0 in 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 for u ε as ε → + 0 . This asymptotic is a sum of a term K ε Ψ ε and a boundary layer, where Ψ ε is the eigenfunction associated with the lowest eigenvalue of the considered problem and K ε is some constant. We provide complete asymptotic expansions for both K ε and Ψ ε ; the boundary layer is also an infinite asymptotic series power in ε . The error term in the asymptotics for u ε is estimated in various norms.

Keywords: exit time problem; equations with small parameter at higher derivatives; asymptotics; overdamped Langevin dynamics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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