Metastability and Time Scales for Parabolic Equations with Drift 1: The First Time Scale

Abstract

Consider the elliptic operator given by

$$\begin{aligned} {\mathscr {L}}_{\varepsilon }f\,=\, {\varvec{b}} \cdot \nabla f \,+\, \varepsilon \, \Delta f \end{aligned}$$

(0.1)

for some smooth vector field \(\varvec{b}:{\mathbb R}^d\rightarrow {\mathbb R}^d\) and a small parameter \(\varepsilon >0\). Consider the initial-valued problem

$$\begin{aligned} \left\{ \begin{aligned}&\partial _ t u_\varepsilon \,=\, {\mathscr {L}}_\varepsilon u_\varepsilon , \\&u_\varepsilon (0, \cdot ) = u_0(\cdot ) , \end{aligned} \right. \end{aligned}$$

(0.2)

for some bounded continuous function \(u_0\). Denote by \(\mathcal {M}_0\) the set of critical points of \(\varvec{b}\) which are stable stationary points for the ODE \(\dot{\varvec{x}} (t) = \varvec{b} (\varvec{x}(t))\). Under the hypothesis that \(\mathcal {M}_0\) is finite and \(\varvec{b} = -(\nabla U + \varvec{\ell })\), where \(\varvec{\ell }\) is a divergence-free field orthogonal to \(\nabla U\), the main result of this article states that there exist a time-scale \(\theta ^{(1)}_\varepsilon \), \(\theta ^{(1)}_\varepsilon \rightarrow \infty \) as \(\varepsilon \rightarrow 0\), and a Markov semigroup \(\{p_t: t\ge 0\}\) defined on \(\mathcal {M}_0\) such that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} u_\varepsilon ( t \, \theta ^{(1)}_\varepsilon , \varvec{x} ) \;=\; \sum _{\varvec{m}'\in \mathcal {M}_0} p_t(\varvec{m}, \varvec{m}')\, u_0(\varvec{m}')\; \end{aligned}$$

for all \(t>0\) and \(\varvec{x}\) in the domain of attraction of \(\varvec{m}\) [for the ODE \(\dot{\varvec{x}}(t) = \varvec{b}(\varvec{x}(t))\)]. The time scale \(\theta ^{(1)}\) is critical in the sense that, for all time scales \(\varrho _\varepsilon \) such that \(\varrho _\varepsilon \rightarrow \infty \), \(\varrho _\varepsilon /\theta ^{(1)}_\varepsilon \rightarrow 0\),

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} u_\varepsilon ( \varrho _\varepsilon , \varvec{x} ) \;=\; u_0(\varvec{m}) \end{aligned}$$

for all \(\varvec{x} \in \mathcal {D}(\varvec{m})\). Namely, \(\theta _\varepsilon ^{(1)}\) is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [20] we extend this result finding all critical time-scales at which the solution of the initial-valued problem (0.2) evolves smoothly in time and we show that the solution \(u_\varepsilon \) is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of \(\varvec{b}\).