TY - JOUR

T1 - Complete asymptotics for solution of singularly perturbed dynamical systems with single well potential

AU - Borisov, Denis I.

AU - Sultanov, Oskar A.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - We consider a singularly perturbed boundary value problem (ε2Δ + ∇V·∇)uε = 0 in Ω, uε = 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 uε as ε → +0. This asymptotic is a sum of a term KεΨε and a boundary layer, where Ψε is the eigenfunction associated with the lowest eigenvalue of the considered problem and Kε is some constant. We provide complete asymptotic expansions for both Kε and Ψε; the boundary layer is also an infinite asymptotic series power in ε. The error term in the asymptotics for uε is estimated in various norms.

AB - We consider a singularly perturbed boundary value problem (ε2Δ + ∇V·∇)uε = 0 in Ω, uε = 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 uε as ε → +0. This asymptotic is a sum of a term KεΨε and a boundary layer, where Ψε is the eigenfunction associated with the lowest eigenvalue of the considered problem and Kε is some constant. We provide complete asymptotic expansions for both Kε and Ψε; the boundary layer is also an infinite asymptotic series power in ε. The error term in the asymptotics for uε is estimated in various norms.

KW - Asymptotics

KW - Equations with small parameter at higher derivatives

KW - Exit time problem

KW - Overdamped langevin dynamics

UR - http://www.scopus.com/inward/record.url?scp=85087561075&partnerID=8YFLogxK

U2 - 10.3390/math8060949

DO - 10.3390/math8060949

M3 - Article

AN - SCOPUS:85087561075

VL - 8

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 6

ER -