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

Consider the elliptic operator given by 0.1 \(\begin{aligned} {\mathscr {L}}_{\varepsilon }f\,=\, {\varvec{b}} \cdot \nabla f \,+\, \varepsilon \, \Delta f \end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{L}}_{\epsilon }f=\mathit{b}·\mathrm{\nabla }f+\epsilon \mathrm{\Delta }f\end{array}$$ for some smooth vector field \(\varvec{b}:{\mathbb R}^d\rightarrow {\mathbb R}^d\) $\mathit{b}:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and a small parameter \(\varepsilon >0\) $\epsilon >0$ . Consider the initial-valued problem 0.2 \(\begin{aligned} \left\{ \begin{aligned}&\partial _ t u_\varepsilon \,=\, {\mathscr {L}}_\varepsilon u_\varepsilon , \\&u_\varepsilon (0, \cdot ) = u_0(\cdot ) , \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\partial }_t{u}_{\epsilon }={\mathcal{L}}_{\epsilon }{u}_{\epsilon },\hfill \\ \hfill & u_{\epsilon }(0,·)=u_0(·),\hfill \end{array}\right)\end{array}$$ for some bounded continuous function \(u_0\) $u_0$ . Denote by \(\mathcal {M}_0\) ${\mathcal{M}}_0$ the set of critical points of \(\varvec{b}\) $\mathit{b}$ which are stable stationary points for the ODE \(\dot{\varvec{x}} (t) = \varvec{b} (\varvec{x}(t))\) $\dot{\mathit{x}}(t)=\mathit{b}(\mathit{x}(t))$ . Under the hypothesis that \(\mathcal {M}_0\) ${\mathcal{M}}_0$ is finite and \(\varvec{b} = -(\nabla U + \varvec{\ell })\) $\mathit{b}=-(\mathrm{\nabla }U+\mathit{\ell })$ , where \(\varvec{\ell }\) $\mathit{\ell }$ is a divergence-free field orthogonal to \(\nabla U\) $\mathrm{\nabla }U$ , the main result of this article states that there exist a time-scale \(\theta ^{(1)}_\varepsilon \) ${\theta }_{\epsilon }^{(1)}$ , \(\theta ^{(1)}_\varepsilon \rightarrow \infty \) ${\theta }_{\epsilon }^{(1)}\to \infty$ as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ , and a Markov semigroup \(\{p_t: t\ge 0\}\) $\{p_t:t\ge 0\}$ defined on \(\mathcal {M}_0\) ${\mathcal{M}}_0$ such that \(\begin{aligned} \lim _{\varepsilon \rightarrow 0} u_\varepsilon ( t \, \theta ^{(1)}_\varepsilon , \varvec{x} ) \;=\; \sum _{\varvec{m}'\in \mathcal {M}_0} p_t(\varvec{m}, \varvec{m}')\, u_0(\varvec{m}')\; \end{aligned}\) $$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim}{u}_{\epsilon }(t{\theta }_{\epsilon }^{(1)},\mathit{x})=\sum _{{\mathit{m}}^{\prime }\in {\mathcal{M}}_0}{p}_t(\mathit{m},{\mathit{m}}^{\prime })u_0({\mathit{m}}^{\prime })\end{array}$$ for all \(t>0\) $t>0$ and \(\varvec{x}\) $\mathit{x}$ in the domain of attraction of \(\varvec{m}\) $\mathit{m}$ [for the ODE \(\dot{\varvec{x}}(t) = \varvec{b}(\varvec{x}(t))\) $\dot{\mathit{x}}(t)=\mathit{b}(\mathit{x}(t))$ ]. The time scale \(\theta ^{(1)}\) ${\theta }^{(1)}$ is critical in the sense that, for all time scales \(\varrho _\varepsilon \) ${\varrho }_{\epsilon }$ such that \(\varrho _\varepsilon \rightarrow \infty \) ${\varrho }_{\epsilon }\to \infty$ , \(\varrho _\varepsilon /\theta ^{(1)}_\varepsilon \rightarrow 0\) ${\varrho }_{\epsilon }/{\theta }_{\epsilon }^{(1)}\to 0$ , \(\begin{aligned} \lim _{\varepsilon \rightarrow 0} u_\varepsilon ( \varrho _\varepsilon , \varvec{x} ) \;=\; u_0(\varvec{m}) \end{aligned}\) $$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim}{u}_{\epsilon }({\varrho }_{\epsilon },\mathit{x})=u_0(\mathit{m})\end{array}$$ for all \(\varvec{x} \in \mathcal {D}(\varvec{m})\) $\mathit{x}\in \mathcal{D}(\mathit{m})$ . Namely, \(\theta _\varepsilon ^{(1)}\) ${\theta }_{\epsilon }^{(1)}$ 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 \(u_\varepsilon \) $u_{\epsilon }$ is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of \(\varvec{b}\) $\mathit{b}$ .