From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics

Consider the one-dimensional elliptic operator given by {equation*} (L_ε f)(x) \;=\; b (x) \, f'(x) \,+\, ε\, a (x)\, f''(x) \;, {equation*} 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 {equation*} \{ {aligned} & ∂ₜ\,uε\,=\,Lε\,uε\;,\\ & uε(0,\,·)=u₀(·)\;, {aligned} .{equation*} 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≤-1, probability measures p(x,·), x∈ R, and kernels Rₜ⁽ᵖ⁾(mⱼ,mₖ), where ⱼ:j∈ Z\ represents the set of stable equilibrium of the ODE ẋ(t) = b(x(t)) such that {equation*} limε→0 uε(tθε⁽ᵖ⁾, x) \;=\;∑j,k∈ Z p(x,m_j)\, Rₜ⁽ᵖ⁾ (m_j,m_k) \,u₀(m_k)\;, {equation*} for all $t>0$ and x∈ R. The solution uε asymptotic behavior description is completed by the characterisation of its behaviour in the intermediate time-scales ε such that ε/θε⁽ᵖ⁾→∞, ε/θε⁽ᵖ⁺¹⁾→0 for some 0≤ p≤, where θε⁽⁰⁾=1, θε(q+1)=+∞. The proof relies on the analysis of the diffusion X_ε(·) induced by the generator L_ε based on the resolvent approach to metastability introduced in [21].