From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics
Abstract
Consider the one-dimensional elliptic operator given by (Lϵf)(x)=b(x)f′(x)+ϵa(x)f″(x), where the drift b:R→R and the diffusion coefficient a:R→R are periodic C1(R) functions satisfying further conditions, and ϵ>0. Consider the initial-valued problem ∂tuϵ=Lϵuϵ,uϵ(0,⋅)=u0(⋅), for some bounded continuous function u0. We prove the existence of time-scales θϵ(1),…,θϵ(q) such that θϵ(1)→∞, θϵ(p+1)∕θϵ(p)→∞, 1≤p≤q−1, probability measures p(x,⋅), x∈R, and kernels Rt(p)(mj,mk), where {mj:j∈Z} represents the set of stable equilibrium of the ODE x˙(t)=b(x(t)) such that limϵ→0uϵ(tθϵ(p),x)=∑j,k∈Zp(x,mj)Rt(p)(m j,mk)u0(mk), for all t>0 and x∈R. The solution uϵ asymptotic behaviour description is completed by the characterisation of its behaviour in the intermediate time-scales ϱϵ such that ϱϵ∕θϵ(p)→∞, ϱϵ∕θϵ(p+1)→0 for some 0≤p≤q, where θϵ(0)=1, θϵ(q+1)=+∞. The proof relies on the analysis of the diffusion Xϵ(⋅) induced by the generator Lϵ based on the resolvent approach to metastability introduced in [21].
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BibTeX
@article{landim2026from,
title = {{From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics}},
author = {Claudio Landim and Christian Maura},
journal = {Electronic Journal of Probability},
year = {2026},
month = sep,
doi = {10.1214/26-ejp1606},
eprint = {2505.20217},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05770845v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics
AU - Claudio Landim
AU - Christian Maura
PY - 2026
DA - 2026/09/30
JO - Electronic Journal of Probability
DO - 10.1214/26-ejp1606
UR - https://hal.science/hal-05770845v1
AB - Consider the one-dimensional elliptic operator given by (Lϵf)(x)=b(x)f′(x)+ϵa(x)f″(x), where the drift b:R→R and the diffusion coefficient a:R→R are periodic C1(R) functions satisfying further conditions, and ϵ>0. Consider the initial-valued problem ∂tuϵ=Lϵuϵ,uϵ(0,⋅)=u0(⋅), for some bounded continuous function u0. We prove the existence of time-scales θϵ(1),…,θϵ(q) such that θϵ(1)→∞, θϵ(p+1)∕θϵ(p)→∞, 1≤p≤q−1, probability measures p(x,⋅), x∈R, and kernels Rt(p)(mj,mk), where {mj:j∈Z} represents the set of stable equilibrium of the ODE x˙(t)=b(x(t)) such that limϵ→0uϵ(tθϵ(p),x)=∑j,k∈Zp(x,mj)Rt(p)(m j,mk)u0(mk), for all t>0 and x∈R. The solution uϵ asymptotic behaviour description is completed by the characterisation of its behaviour in the intermediate time-scales ϱϵ such that ϱϵ∕θϵ(p)→∞, ϱϵ∕θϵ(p+1)→0 for some 0≤p≤q, where θϵ(0)=1, θϵ(q+1)=+∞. The proof relies on the analysis of the diffusion Xϵ(⋅) induced by the generator Lϵ based on the resolvent approach to metastability introduced in [21].
SN - 1083-6489
ER -