2.1. $\varepsilon$ -Fokker–Planck equation (2.4) as a Kolmogorov equation

Consider a drift-diffusion process $(X_t)_{t\geq 0}$ , described by the following stochastic differential equation

(2.5) \begin{equation} \,\mathrm{d} X_t = b(X_t) \,\mathrm{d} t + \sigma (X_t) * \,\mathrm{d} B_t, \end{equation}

where $B_t$ is a one-dimensional Brownian motion, and

(2.6) \begin{equation} b(x) = - {B_\varepsilon ^{-1}}(x) \nabla U_\varepsilon (x), \quad \text{and}\quad \sigma (x) = \sqrt {2 {B_\varepsilon ^{-1}}(x)}. \end{equation}

Here the multiplicative noise $\sigma (X_t) * \,\mathrm{d} B_t$ is in the backward Ito differential sense, which is equivalent to the forward Ito differential by adding an additional drift term

\begin{equation*} \sigma (X_t)* \,\mathrm{d} B_t = \frac 12\nabla \cdot (\sigma \sigma ^T)(X_t) \,\mathrm{d} t + \sigma (X_t) \,\mathrm{d} B_t. \end{equation*}

By Ito’s formula, the generator of the process $(X_t)_{t\geq 0}$ is derived as follows. For any test function $\varphi \in C^2_b(\mathbb{R}^n)$ and initial condition $X_0=x$ , we compute

(2.7) \begin{equation} \begin{aligned} \lim _{t\to 0^+} \frac {{\mathbb{E}}^x[\varphi (X_t)]-\varphi (x)}{t} = &\lim _{t\to 0^+} {\mathbb{E}}^x \frac {1}{t}\int _0^t \big [ \nabla \varphi (X_s) \cdot b(X_s) \\ &+ \frac 12 \nabla ^2 \varphi (X_s) \,:\, (\sigma \sigma ^T)(X_s) + \frac 12 (\nabla \cdot (\sigma \sigma ^T)(X_s)) \cdot \nabla \varphi (X_s) \big ] \,\mathrm{d} s\\ =& \nabla \varphi (x) \cdot b(x) + \frac 12 \nabla \cdot (\sigma \sigma ^T \nabla \varphi (x))\,=\!:\, \mathcal{L}\varphi . \end{aligned} \end{equation}

Thus the corresponding Fokker–Planck equation to (2.5) is given by

(2.8) \begin{eqnarray} \partial _t \rho _t^\varepsilon &=& \mathcal{L}^*\rho ^\varepsilon _t \nonumber \\ &\,:\!=\,& \frac 12 \nabla \cdot \left (\sigma \sigma ^T \nabla \rho _t^\varepsilon \right ) -\nabla \cdot \left (\rho _t^\varepsilon b\right )\nonumber \\ & = & \nabla \cdot \left ({B_\varepsilon ^{-1}}(x) \nabla \rho _t^\varepsilon (x)\right ) + \nabla \cdot (\rho _t^\varepsilon (x) {B_\varepsilon ^{-1}}(x) \nabla U_\varepsilon (x) ), \end{eqnarray}

which is exactly (2.4). Note that the $\pi _\varepsilon$ defined in (2.2), which is in the form of a Gibbs measure, is in fact the unique stationary distribution of (2.8), $\mathcal{L}^*\pi _\varepsilon = 0$ .

We remark that in the above drift-diffusion process, we used the Ito backward differential to ensure that our process $(X_t)_{t\geq 0}$ with a multiplicative noise is reversible so that one can have a gradient flow structure for the corresponding Fokker–Planck equation. More precisely, we have that the diffusion process $(X_t)_{t\geq 0}$ (2.5) starting from $X_0 \sim \pi _\varepsilon$ is reversible in the sense that the time reversed process has the same distribution, i.e.

(2.9) \begin{equation} {\mathbb{E}}(\varphi _1(X_t)\varphi _2(X_0)|X_0\sim \pi _\varepsilon ) = {\mathbb{E}}(\varphi _1(X_0)\varphi _2(X_t)|X_0\sim \pi _\varepsilon ), \quad \forall \varphi _1,\varphi _2\in C_0^\infty (\mathbb{R}^n),\, \forall t\gt 0. \end{equation}

This condition is equivalent to the symmetry of the generator $\mathcal{L}$ in $L^2(\pi _\varepsilon )$ ; cf. [Reference Gao, Liu and Liu25].

2.2. $\varepsilon$ -Fokker-Planck equation (2.4) as a gradient flow in $(\mathcal{P}(\Omega ), W_\varepsilon )$

Following Otto’s formal Riemannian calculus on Wasserstein space [Reference Otto40], we now interpret the Fokker–Planck equation as a (negative) gradient flow in $(\mathcal{P}(\Omega ), W_\varepsilon )$ . For this purpose, we need to compute the Wasserstein gradient $\nabla ^{W_\varepsilon } E_\varepsilon$ of $E_\varepsilon$ in $(\mathcal{P}(\Omega ), W_\varepsilon )$ .

Given any absolutely continuous curve $\tilde {\rho }_t$ in $(\mathcal{P}(\Omega ), W_\varepsilon )$ given by $\tilde {\rho }_t \,:\!=\, (\chi _t)_\# \rho$ with $\tilde {\rho }_{t=0}=\rho$ , where $\chi _t$ is the flow map induced by a smooth velocity field $v_t$ . Then $\tilde {\rho }_t$ satisfies the continuity equation

\begin{equation*}\partial _t \tilde {\rho }_t + \nabla \cdot \left (\tilde {\rho }_t v_t \right )=0.\end{equation*}

With this, we compute the first variation of $E_\varepsilon$

(2.10) \begin{equation} \frac {\,\mathrm{d}}{\,\mathrm{d} t}\Big |_{t=0} E_\varepsilon (\tilde {\rho }_t) = \int _\Omega \frac {\delta E_\varepsilon }{\delta \rho } \partial _t \tilde {\rho }_t\big |_{t=0} \,\mathrm{d} x = \int _\Omega \frac {\delta E_\varepsilon }{\delta \rho } \left ( -\nabla \cdot (\tilde {\rho }_tv_t)\big |_{t=0} \right )\,\mathrm{d} x = \int _\Omega \left \langle \nabla \frac {\delta E_\varepsilon }{\delta \rho }, v_0 \right \rangle \rho \,\mathrm{d} x. \end{equation}

We will use the above to identify the gradient $\nabla ^{W_\varepsilon } E_\varepsilon$ of $E_\varepsilon$ with respect to a Riemannian metric $\langle \cdot , \cdot \rangle _{T_{\mathcal{P}}, T_{\mathcal{P}}}$ on the tangent plane $T_{\mathcal{P}}$ of $(\mathcal{P}(\Omega ), W_\varepsilon )$ .

Based on (1.5), we have that for any $\rho \in \mathcal{P}(\Omega )$ and $s_1, s_2\in T_{\mathcal{P}}$ at $\rho$ , the metric is given by

(2.11) \begin{equation} \big \langle s_1, s_2\big \rangle _{T_{\mathcal{P}}, T_{\mathcal{P}}}\,:\!=\, \int \rho (x)\left \langle B^{-1}_\varepsilon (x) \nabla \varphi _1(x), \nabla \varphi _2(x)\right \rangle \,dx, \quad \text{ where } s_i = -\nabla \cdot (\rho B^{-1}_\varepsilon \nabla \varphi _i),\,\,\,i=1,2. \end{equation}

A word is in place here to explain going from $v_t$ in (1.5) to $\nabla \varphi$ above. At a fixed $t$ and $\rho _t$ , upon minimising $\displaystyle \int _\Omega \rho _t\langle B_\varepsilon (x) v_t, v_t\rangle \,\,\mathrm{d} x$ over $v_t$ subject to $\displaystyle - \nabla \cdot (\rho _t v_t) = s \left (\,:\!=\,\frac {\partial \rho _t}{\partial t}\right )$ , we have that

\begin{equation*} \int _\Omega \rho _t\langle B_\varepsilon (x) v_t, \xi \rangle \,\,\mathrm{d} x = 0 \,\,\,\text{for all smooth vector field $\xi $ satisfying $- \nabla \cdot (\rho _t \xi ) = 0$}. \end{equation*}

Hence, $B_\varepsilon v_t$ is orthogonal to all divergence free vector field of the form $\rho _t\xi$ . We then conclude that $B_\varepsilon v_t$ must be the gradient of some (potential) function $\varphi$ . Thus, $v_t$ can be represented as $v_t = B^{-1}_\varepsilon \nabla \varphi$ .

With the above, we express the first variation of $E_\varepsilon$ using $\nabla ^{W_\varepsilon }E_\varepsilon$ as follows:

(2.12) \begin{equation} \frac {\,\mathrm{d}}{\,\mathrm{d} t}\Big |_{t=0} E_\varepsilon (\tilde {\rho }_t) = \Big \langle \nabla ^{W_\varepsilon } E_\varepsilon , \partial _t \tilde {\rho }_t\big |_{t=0} \Big \rangle _{T_{\mathcal{P}}, T_{\mathcal{P}}} = \int _\Omega \rho \langle B^{-1}_\varepsilon \nabla \tilde {\varphi } , \nabla \varphi _0 \rangle \,\mathrm{d} x, \end{equation}

where

(2.13) \begin{equation} \partial _t\tilde {\rho }_t\big |_{t=0} = -\nabla \cdot \left ( \rho B^{-1}_\varepsilon \nabla \varphi _0 \right ) \quad \text{and}\quad \nabla ^{W_\varepsilon } E_\varepsilon (\rho )= -\nabla \cdot \left (\rho B^{-1}_\varepsilon \nabla \tilde {\varphi } \right ). \end{equation}

Comparing (2.10) with (2.12), we have

\begin{equation*} \int _\Omega \left \langle \nabla \frac {\delta E_\varepsilon }{\delta \rho }, v_0 \right \rangle \rho \,\mathrm{d} x = \int _\Omega \rho \big \langle B^{-1}_\varepsilon \nabla \tilde {\varphi } , \nabla \varphi _0 \big \rangle \,\mathrm{d} x \end{equation*}

which is set to hold for any $v_0=B^{-1}_\varepsilon \nabla \varphi _0$ . Hence, $\displaystyle \nabla \tilde {\varphi } = \nabla \frac {\delta E_\varepsilon }{\delta \rho }$ . Thus, the second part of (2.13) leads to the following identification of $\nabla ^{W_\varepsilon } E (\rho )$ ,

(2.14) \begin{equation} \nabla ^{W_\varepsilon } E_\varepsilon (\rho ) \,:\!=\, -\nabla \cdot \left (\rho B_\varepsilon ^{-1} \nabla \frac {\delta E_\varepsilon }{\delta \rho }\right )=-\nabla \cdot \left (\rho B_\varepsilon ^{-1} \nabla \log \frac {\rho }{\pi _\varepsilon }\right ). \end{equation}

Hence, the inhomogeneous Fokker–Planck equation (2.4) indeed can be written as a gradient flow of $E_\varepsilon$ with respect to the $\varepsilon$ -Wasserstein metric $W_\varepsilon$ , i.e.,

(2.15) \begin{equation} \partial _t \rho ^\varepsilon _t = -\nabla ^{W_\varepsilon } E_\varepsilon (\rho ^\varepsilon _t) = \nabla \cdot \left (\rho ^\varepsilon B_\varepsilon ^{-1} \nabla \log \frac {\rho ^\varepsilon }{\pi _\varepsilon }\right ). \end{equation}

We remark that in general an equation may have many different gradient flow structures with respect to the same free energy $E_\varepsilon$ , cf. [Reference Mielke, Montefusco and Peletier38]. However, in this paper, we restrict ourselves within the framework of Wasserstein gradient flows as it fits naturally to the evolution in probability space.

2.3. $\varepsilon$ -generalised gradient flow in energy-dissipation inequality (EDI) form

As mentioned previously, in order to study the limiting gradient flow structure as the small parameter $\varepsilon \to 0$ in our $\varepsilon$ -gradient flow (2.15), we will recast it in an energy-dissipation inequality (EDI) form (1.13) that is shown to be equivalent to the original $\varepsilon$ -gradient flow system.

Denote the $\varepsilon$ -dissipation on the tangent plane $T_{\mathcal{P}}$ as a functional $\psi _\varepsilon \,:\,\mathcal{P} \times T_{\mathcal{P}} \to \mathbb{R}$ defined by

(2.16) \begin{equation} \psi _\varepsilon (\rho , s)\,:\!=\, \frac 12 \int _\Omega \langle \nabla u, B_\varepsilon ^{-1} \nabla u \rangle \rho \,\mathrm{d} x, \quad \text{ with }\, s=-\nabla \cdot \left (\rho B_\varepsilon ^{-1} \nabla u\right ), \end{equation}

and the $\varepsilon$ -dissipation on the cotangent plane $T^*_{\mathcal{P}}$ as a functional $\psi ^*_\varepsilon \,:\, \mathcal{P} \times T^*_{\mathcal{P}} \to \mathbb{R}$ defined by

(2.17) \begin{equation} \psi ^*_\varepsilon (\rho , \xi )\,:\!=\, \frac 12 \int _\Omega \langle \nabla \xi , B_\varepsilon ^{-1} \nabla \xi \rangle \rho \,\mathrm{d} x. \end{equation}

It is easy to check that

(2.18) \begin{equation} \begin{aligned} \psi _\varepsilon (\rho ,s) =& \sup _{\xi \in T^*_\rho } \Big \{ \langle \xi , s \rangle _{T^*_\rho , T_\rho } - \psi ^*_\varepsilon (\rho , \xi ) \Big \}\\ =& \langle \xi ^*, s \rangle _{T^*_\rho , T_\rho } - \psi ^*_\varepsilon (\rho , \xi ^*) \quad \text{ with } s= -\nabla \cdot \left (\rho B_\varepsilon ^{-1}\nabla \xi ^* \right )\\ =& \frac 12 \int _\Omega \langle \nabla \xi ^*, B_\varepsilon ^{-1} \nabla \xi ^* \rangle \rho \,\mathrm{d} x. \end{aligned} \end{equation}

Applying the Fenchel–Young inequality to the convex functionals $\psi _\varepsilon$ and $\psi ^*_\varepsilon$ , we have

(2.19) \begin{equation} \langle \xi , s\rangle \leq \psi _\varepsilon ^*(\rho , \xi ) + \psi _\varepsilon (\rho , s), \quad \text{for all}\quad \xi \in T_\rho ^*,\,\,\,\text{and}\,\,\,s\in T_\rho , \end{equation}

with equality holds if and only if $\xi \in \partial _s\psi _\varepsilon (\rho ,s)$ and $s\in \partial _\xi \psi ^*_\varepsilon (\rho ,\xi )$ . Here $\partial _s\psi _\varepsilon (\rho , s)$ and $\partial _\xi \psi _\varepsilon ^*(\rho , \xi )$ refer to the subdifferentials of $\psi _\varepsilon$ and $\psi _\varepsilon ^*$ on $T_\rho$ and $T_\rho ^*$ , respectively, at a fixed $\rho$ . We also note the following.

1. (1) For all $\eta \in T^*_{\mathcal{P}}$ , we have

   \begin{eqnarray*} &&\Big \langle \partial _\xi \psi ^*_\varepsilon (\rho , \xi ),\eta \Big \rangle = \lim _{\tau \to 0}\left . \frac {d}{d\tau }\psi ^*_\varepsilon (\rho , \xi +\tau \eta )\right |_{\tau =0}\\ &=& \int \langle \nabla \xi , B_\varepsilon ^{-1}\nabla \eta \rangle \rho \,\,\mathrm{d} x =\int -\eta \nabla \cdot (\rho B_\varepsilon ^{-1}\nabla \xi )\,\,\mathrm{d} x \end{eqnarray*}
   so that $\partial _\xi \psi ^*_\varepsilon (\rho ,\xi )=-\nabla \cdot (\rho B_\varepsilon ^{-1}\nabla \xi )$ . Hence, $s\in \partial _\xi \psi ^*_\varepsilon (\rho , \xi )$ means $s=-\nabla \cdot (\rho B_\varepsilon ^{-1}\nabla \xi )$ .

2. (2) For all $\sigma \in T_{\mathcal{P}}$ , we have

   \begin{eqnarray*} &&\Big \langle \partial _s\psi _\varepsilon (\rho , s),\sigma \Big \rangle = \lim _{\tau \to 0}\left . \frac {d}{d\tau }\psi _\varepsilon (\rho , s+\tau \sigma )\right |_{\tau =0}\\ &=& \int \langle \nabla u, B_\varepsilon ^{-1}\nabla \omega \rangle \rho \,\,\mathrm{d} x = \int -u\nabla \cdot (\rho B_\varepsilon ^{-1}\nabla \omega )\,\,\mathrm{d} x\\ && \Big (\text{where}\,\,\, s=-\nabla \cdot (\rho B_\varepsilon ^{-1}\nabla u),\,\,\, \sigma =-\nabla \cdot (\rho B_\varepsilon ^{-1}\nabla \omega ) \Big ) \\ &=& \int u\sigma \,\,\mathrm{d} x \end{eqnarray*}
   so that $\partial _s\psi _\varepsilon (\rho ,s)=u$ . Hence, $\xi \in \partial _s\psi _\varepsilon (\rho , s)$ means $\xi$ satisfies $s=-\nabla \cdot (\rho B_\varepsilon ^{-1}\nabla \xi )$ .

With the above, we now reformulate (2.15) in the form of an EDI. To this end, we compute,

(2.20) \begin{equation} \frac {d}{dt}E_\varepsilon (\rho _t^\varepsilon ) = \left \langle \frac {\delta E_\varepsilon }{\delta \rho },\partial _t\rho _t^\varepsilon \right \rangle , \,\,\,\text{or}\,\,\, \frac {d}{dt}E_\varepsilon (\rho _t^\varepsilon ) + \left \langle -\frac {\delta E_\varepsilon }{\delta \rho },\partial _t\rho _t^\varepsilon \right \rangle = 0. \end{equation}

By (2.19), $\partial _t\rho _t^\varepsilon = -\nabla ^{W_\varepsilon }E_\varepsilon (\rho _t^\varepsilon )=\nabla \cdot \left (\rho B_\varepsilon ^{-1} \nabla \frac {\delta E_\varepsilon }{\delta \rho }\right )$ if and only if

\begin{equation*} \psi _\varepsilon (\rho ^\varepsilon _\tau , \partial _\tau \rho ^\varepsilon _\tau ) + \psi ^*_\varepsilon \left (\rho _\tau ^\varepsilon , -\frac {\delta E_\varepsilon }{\delta \rho }(\rho _\tau ^\varepsilon )\right ) \leq \left \langle -\frac {\delta E_\varepsilon }{\delta \rho },\partial _t\rho _t^\varepsilon \right \rangle . \end{equation*}

Hence, upon integrating (2.20), our gradient flow (2.15) is equivalent to the following:

(2.21) \begin{equation} E_\varepsilon (\rho _t^\varepsilon ) + \int _0^t \left [ \psi _\varepsilon (\rho ^\varepsilon _\tau , \partial _\tau \rho ^\varepsilon _\tau ) + \psi ^*_\varepsilon \left (\rho _\tau ^\varepsilon , -\frac {\delta E_\varepsilon }{\delta \rho }(\rho _\tau ^\varepsilon )\right ) \right ] \,\mathrm{d} \tau \leq E_\varepsilon (\rho _0^\varepsilon ). \end{equation}

We note that the very first step, (2.20) is a crucial chain rule of differentiation. This is justified in our paper due to the regularity property of our energy functional and the solution. Precise statements will be given in Section 3. In general (for example, discrete or general metric space) settings, the absolute continuity of $E_\varepsilon (\rho _t^\varepsilon )$ (in time) and the validity of the chain rule (2.20) need to be proved; cf., [Reference Hoeksema and Tse29, Reference Hraivoronska, Schlichting and Tse28].

Before leaving this section, for convenience, we write down the following explicit expressions.

(2.22) \begin{eqnarray} \psi ^*_\varepsilon \left (\rho _\tau ^\varepsilon , -\frac {\delta E_\varepsilon }{\delta \rho }(\rho _\tau ^\varepsilon )\right ) &=& \int _\Omega \left \langle \nabla \left (\frac {\delta E_\varepsilon }{\delta \rho }\right ), B_\varepsilon ^{-1} \nabla \left (\frac {\delta E_\varepsilon }{\delta \rho }\right ) \right \rangle \rho _t^\varepsilon \,\,\mathrm{d} x\nonumber \\ &=& \frac 12 \int _\Omega \left \langle \nabla \log \frac {\rho ^\varepsilon _\tau }{\pi _\varepsilon }, {B_\varepsilon ^{-1}} \nabla \log \frac {\rho ^\varepsilon _\tau }{\pi _\varepsilon } \right \rangle \rho ^\varepsilon _\tau \,\mathrm{d} x, \end{eqnarray}

and

(2.23) \begin{equation} \psi _\varepsilon \left (\rho ^\varepsilon _\tau , \partial _\tau \rho ^\varepsilon _\tau \right ) = \frac 12 \int _\Omega \langle \nabla u, B_\varepsilon ^{-1} \nabla u \rangle \rho _\tau ^\varepsilon \,\mathrm{d} x, \quad \text{ with }\, -\nabla \cdot \left (\rho ^\varepsilon _\tau B_\varepsilon ^{-1} \nabla u\right ) = \partial _\tau \rho _\tau ^\varepsilon . \end{equation}

2.4. Main results

Briefly stated, our main result is that the gradient-flow structure is preserved in the limit, i.e., (2.15) converges to a limiting gradient flow. More precisely, the solution $\rho ^\varepsilon _t$ of (2.15) converges (weakly) to $\rho _t$ that solves a gradient flow with respect to a limiting Wasserstein distance $\overline {W}$ ,

(2.24) \begin{equation} \partial _t \rho _t = -\nabla ^{\overline {W}}\overline {E}(\rho _t) = \nabla \cdot \left (\rho _t \overline {B}^{-1} \nabla \log \frac {\rho _t}{\overline {\pi }}\right ). \end{equation}

In the above, the limiting energy is given as

(2.25) \begin{equation} \overline {E}(\rho ) = \text{KL}(\rho ||\overline {\pi }) = \int _\Omega \rho \log \frac {\rho }{\overline {\pi }}\,\,\mathrm{d} x, \end{equation}

where the $\overline {\pi }$ is simply the spatial average of $\pi _\varepsilon$ with respect to some fast variable – see (2.33) below. The matrix $\overline {B}$ is obtained by taking appropriate average of $B_\varepsilon$ over the fast variable weighted by the solution of a cell problem (A.9) or equivalently, by considering the $\Gamma$ -limit of a variational functional (Theorem4.2). The Wasserstein distance $\overline {W}$ is related to $\overline {B}$ just as the way $W_\varepsilon$ is related to $B_\varepsilon$ – see Section 5.1.

Similar to (2.21), (2.16), and (2.17), equation (2.24) is formulated as an EDI, i.e.,

(2.26) \begin{equation} \overline {E}(\rho _t) + \int _0^t \left [ \psi (\rho _\tau , \partial _\tau \rho _\tau ) + \psi ^*\left (\rho _\tau , -\frac {\delta \overline {E}}{\delta \rho }(\rho _\tau )\right ) \right ] \,\mathrm{d} \tau \leq \overline {E}(\rho _0), \end{equation}

where $\psi ^{*}\,:\, \mathcal{P} \times T^*_{\mathcal{P}} \to \mathbb{R}$ is the limiting dissipation functional on the cotangent plane $T^*_{\mathcal{P}}$ given by

(2.27) \begin{equation} \psi ^*(\rho , \xi )\,:\!=\, \frac 12 \int _\Omega \langle \nabla \xi , \bar {B}^{-1} \nabla \xi \rangle \rho \,\mathrm{d} x, \end{equation}

and $\psi \,:\, \mathcal{P} \times T_{\mathcal{P}} \to \mathbb{R}$ is the limiting dissipation functional on the tangent plane $T_{\mathcal{P}}$ given by

(2.28) \begin{equation} \psi (\rho , s)\,:\!=\, \frac 12 \int _\Omega \langle \nabla u, \bar {B}^{-1} \nabla u \rangle \rho \,\mathrm{d} x, \quad \text{ with }\, s=-\nabla \cdot \left (\rho \bar {B}^{-1} \nabla u\right ). \end{equation}

The precise statement of the convergence of (2.21) to (2.26) will be given in Section 4, Theorem4.1.

Curiously, under the current setting, $\overline {W}$ is not the Gromov–Hausdorff limit $W_{\text{GH}}$ of $W_\varepsilon$ which is the common mode of convergence for metric spaces, cf. [Reference Villani53, Reference Gigli and Maas26, Reference Gladbach, Kopfer and Maas21]. In Section 5.2, We have constructed examples such that $\overline {W}$ is strictly bigger than $W_{\text{GH}}$ . We believe that this statement is true for general heterogeneous media.

Before proceeding further, we introduce the following notations and conventions. As we will often consider functions that oscillate on a small length scale, $0 \lt \varepsilon \ll 1$ , it is convenient to introduce the following fast variable

(2.29) \begin{equation} y=\frac {x}{\varepsilon }. \end{equation}

The domain for $y$ is taken to be the $n$ -dimensional torus $\mathbb T^n$ when the oscillatory functions are $1$ -periodic in $y$ . The notation $\overline {A}$ means that it is derived from some averaging of $A$ over the fast variable $y$ . For time-dependent problems, we often deal with functions defined on both space and time variables $x,t$ . For ease of notation, given a function $f=f(x,t)$ , we often use $f_t$ to denote $f_t(\cdot )$ , i.e., the slice of $f$ at a fixed time $t$ . We will use $\rightharpoonup$ and $\longrightarrow$ to denote weak and strong convergence in some function spaces. Two common spaces used are the space of probability measures $\mathcal{P}(\Omega )$ and $L^p(\Omega )$ spaces. The value of $p$ will depend on contexts. For the convergence of a sequence of functions $f_\varepsilon$ as $\varepsilon \to 0$ , we will use the same notation even if the convergence only holds upon extraction of subsequence. (The convergence can be established for the whole sequence if the limiting equation has unique solution which is the case for our linear Fokker–Planck equation (2.24).)

Next we state the main assumptions for our results. Some of these are made only for simplicity. They can be relaxed if we choose to use more technical tools.

1. (i) Recall that the domain $\Omega$ is taken to be an $n$ -dimensional torus ${\mathbb T}^n$ . This is not to be confused with the ${\mathbb T}^n$ for the fast variable $y$ . We note that the boundedness of the domain can be removed, allowing one to work in $P_2(\mathbb{R}^n)$ if a confinement potential $U$ is incorporated in the dynamics. Other boundary conditions, such as Dirichlet or no-flux conditions, may also be considered.

2. (ii) For $B_\varepsilon$ , we consider

   (2.30) \begin{equation} B_\varepsilon (x) = B\left (\frac {x}{\varepsilon }\right ),\,\,\,\text{or}\,\,\, B_\varepsilon (x) = B(y), \end{equation}
   where $B(\cdot )$ is $1$ -periodic. Furthermore, $B(\cdot )$ is bounded and uniformly positive definite, i.e., there are $C_1, C_2 \gt 0$ such that for all $y\in \mathbb T^n$ , it holds that
   (2.31) \begin{equation} C_1 I \leq B(y) \leq C_2 I. \end{equation}
   This form of $B_\varepsilon$ can certainly be generalised to allow for dependence on the slow variable: $B_\varepsilon (x)=B(x,\frac {x}\varepsilon )$ . For simplicity, we assume further that $B$ is smooth in $y$ .

3. (iii) For $\pi _\varepsilon$ , we consider the following form of separation of length scales:

   (2.32) \begin{equation} \pi _\varepsilon (x) = \pi \left (x,\frac {x}{\varepsilon }\right ). \end{equation}
   In the above, $\pi$ is $1$ -periodic in the fast variable $\displaystyle y=\frac {x}{\varepsilon }$ . We further assume that $\pi$ is smooth in both $x$ and $y$ and is bounded away from zero and from above uniformly in $\varepsilon \gt 0$ . The following notation referring to an averaged version of $\pi$ will be used in this paper:
   (2.33) \begin{equation} \overline {\pi }(x) = \int \pi (x,y)\,\mathrm{d} y. \end{equation}
   As concrete examples, $\pi _\varepsilon$ can be taken as
   (2.34) \begin{equation} \pi _\varepsilon ^{\text{I}}(x) = \pi _0(x) + \pi _1\left (x,\frac {x}{\varepsilon }\right ), \quad \text{or}\quad \pi _\varepsilon ^{\text{II}}(x) = \pi _0(x) + \varepsilon \pi _1\left (x,\frac {x}{\varepsilon }\right ). \end{equation}
   Then $\pi _\varepsilon ^{\text{I}}$ and $\pi _\varepsilon ^{\text{II}}$ converge as follow:
   (2.35) \begin{equation} \pi _\varepsilon ^{\text{I}}(x) \rightharpoonup \overline {\pi }^{\text{I}}(x) \,:\!=\, \pi _0(x) + \int _{\mathbb T^n}\pi _1(x,y)\,\,\mathrm{d} y ,\,\,\,\text{and}\,\,\, \pi _\varepsilon ^{\text{II}}(x) \longrightarrow \overline {\pi }^{\text{II}}(x) \,:\!=\, \pi _0(x). \end{equation}
   We thus call $\pi _\varepsilon ^{\text{I}}$ the oscillatory case while $\pi _\varepsilon ^{\text{II}}$ the uniform case. (We refer to the work [Reference Dupuis and Spiliopoulos19] for large deviations for multiscale diffusion with $\pi _\varepsilon ^{\text{II}}$ .)

4. (iv) The initial data $\rho ^\varepsilon _0$ is bounded away from zero and from above uniformly in $\varepsilon \gt 0$ . It is assumed to be well-prepared in the following sense,

   (2.36) \begin{equation} \text{there is a $\rho _0$ such that as $\varepsilon \to 0$, it holds $\rho ^\varepsilon _0 \rightharpoonup \rho _0$ and $E_\varepsilon (\rho ^\varepsilon _0) \to \overline {E}(\rho _0), \quad \text{ as } \, \varepsilon \to 0,$} \end{equation}
   where $E_\varepsilon$ and $\overline {E}$ are given by (2.1) and (2.25). More precise smoothness requirements on $\rho _0$ will be listed in Lemmas3.1, 3.3, and Corollaries3.2 and 3.4.

We have the following remarks about our results.

5.1. Effective Wasserstein distance $\overline {W}$ induced by convergence of gradient flows

For convenience, we recall here the Kantorovich and Benamou–Brenier formulations (1.4) and (1.5) for our $\varepsilon$ -Wasserstein metric $W_\varepsilon$ :

(5.2) \begin{equation} W_\varepsilon ^2(\rho _0, \rho _1)\,:\!=\,\inf \left \{ \iint d^2_\varepsilon (x,y) \,\mathrm{d} \gamma (x,y); \quad \int _{\Omega } \gamma ( x,\,\mathrm{d} y) = \rho _0(x)\,\mathrm{d} x, \,\, \int _{\Omega } \gamma (\,\mathrm{d} x, y) = \rho _1(y)\,\mathrm{d} y \right \} \end{equation}

and

(5.3) \begin{equation} W_\varepsilon ^2(\rho _0, \rho _1) \,:\!=\, \inf \left \{ \int _0^1\int \rho _t(x)\langle B_\varepsilon (x) v_t(x), v_t(x)\rangle \,\mathrm{d} x\,\mathrm{d} t,\quad (\rho _t, v_t)\in V(\rho _0, \rho _1) \right \}, \end{equation}

where $V$ is defined in (1.6). The $\varepsilon$ -metric $d_\varepsilon$ on $\Omega \subset \mathbb{R}^n$ is given via the least action

(5.4) \begin{equation} d_\varepsilon ^2(x,y) \,:\!=\, \inf \left \{ \int _0^1 \langle B_\varepsilon (z_t) \dot {z}_t, \dot {z}_t \rangle \,\mathrm{d} t, \quad z_0 = x, \quad z_1=y \right \}. \end{equation}

A curve $z(\cdot )\in AC([0,1]; \mathbb{R}^n)$ that achieves the infimum in (5.4) is a geodesic in the metric space $(\mathbb{R}^n, d_\varepsilon ).$ From [Reference Bernard and Buffoni6, Theorem A,B], (5.2) and (5.3) are equivalent.

The same formulations hold for our induced limit Wasserstein distance $\overline {W}$ . More precisely, we have

(5.5) \begin{equation} \overline {W}^2(\rho _0, \rho _1)\,:\!=\, \inf \left \{ \int \int \overline {d}^2(x,y) \,\mathrm{d} \gamma (x,y); \quad \int _{\Omega } \gamma ( x,\,\mathrm{d} y) = \rho _0(x)\,\mathrm{d} x, \,\, \int _{\Omega } \gamma (\,\mathrm{d} x, y) = \rho _1(y)\,\mathrm{d} y \right \}, \end{equation}

and the equivalent formulation

(5.6) \begin{equation} \overline {W}^2(\rho _0, \rho _1)\,:\!=\, \inf \left \{ \int _0^1\int \rho _t(x)\langle \overline {B} v_t(x), v_t(x)\rangle \,\mathrm{d} x\,\mathrm{d} t,\quad (\rho _t, v_t)\in V(\rho _0, \rho _1) \right \}. \end{equation}

Here the constant matrix $\overline {B}$ is defined in (A.9) and the induced-metric $\overline {d}$ on $\Omega \subset \mathbb{R}^n$ is again given via the least action

(5.7) \begin{equation} \overline {d}^2(x,y) \,:\!=\, \inf \left \{ \int _0^1 \langle \overline {B} \dot {z}_t, \dot {z}_t \rangle \,\mathrm{d} t, \quad z_0 = x, \quad z_1=y \right \}. \end{equation}

From the Euler–Lagrangian equation for the minimiser of (5.7), the optimal curve $\tilde {z}(\cdot )$ that achieves the least action satisfies $\overline {B}\ddot {\tilde {z}}_t=0$ , and hence it has constant speed, $\dot {\tilde {z}}_t = y-x$ . Thus, we have explicitly

(5.8) \begin{equation} \overline {d}^2(x,y) = \langle \overline {B} (y-x), y-x \rangle = \langle \overline {B} \hat {n}, \hat {n} \rangle |y-x|^2, \quad \text{where}\quad \hat {n}=\frac {y-x}{|y-x|}. \end{equation}

Note that both $W^\varepsilon$ and $\overline {W}$ induce a Riemannian metric on $\mathcal{P}(\Omega )$ . More precisely, for any $\rho \in \mathcal{P}(\Omega )$ , and any $s_1, s_2\in T_{\mathcal{P}}$ , the tangent plane at $\rho$ , the first fundamental form are defined, respectively, as

(5.9) \begin{eqnarray} \big \langle s_1, s_2\big \rangle _{T_{\mathcal{P}}, T_{\mathcal{P}}, \varepsilon }&\,:\!=\,& \int \rho (x)\langle B^{-1}_\varepsilon (x) \nabla u_1(x), \nabla u_2(x)\rangle \,\mathrm{d} x, \end{eqnarray}

where $s_i = -\nabla \cdot (\rho B^{-1}_\varepsilon \nabla u_i)$ , $i=1,2$ for $W^\varepsilon$ , and

(5.10) \begin{eqnarray} \big \langle s_1, s_2\big \rangle _{T_{\mathcal{P}}, T_{\mathcal{P}}}&\,:\!=\,& \int \rho (x)\langle \overline {B}^{-1}(x) \nabla u_1(x), \nabla u_2(x)\rangle \,\mathrm{d} x, \end{eqnarray}

where $s_i = -\nabla \cdot (\rho \overline {B}^{-1}\nabla u_i)$ , $i=1,2$ for $\overline {W}$ . This is also manifested by the fact that both the corresponding dissipation functionals are bilinear forms in $s$ :

\begin{equation*} \psi _\varepsilon (\rho , s)= \frac 12 \int _\Omega \langle \nabla u, B_\varepsilon ^{-1} \nabla u \rangle \rho \,\mathrm{d} x \quad \text{ with }\, s=-\nabla \cdot \left (\rho B_\varepsilon ^{-1} \nabla u\right ), \end{equation*}

and

\begin{equation*} \psi (\rho , s)= \frac 12 \int _\Omega \langle \nabla u, \overline {B}^{-1} \nabla u \rangle \rho \,\mathrm{d} x \quad \text{ with }\, s=-\nabla \cdot \left (\rho \overline {B}^{-1} \nabla u\right ). \end{equation*}

5.2. The Gromov–Hausdorff limit $W_{\text{GH}}$ of $W_\varepsilon$

Now we consider the convergence in the Gromov–Hausdorff sense of $W_\varepsilon$ to a limiting Wasserstein metric, denoted as $W_{\text{GH}}$ .

We first show that even in one dimension, in general it is always the case that $W_{\text{GH}}\lt \overline {W}$ unless $\pi _\varepsilon$ and $B_\varepsilon$ are related to each other in some specific way. Recall the metric $d_\varepsilon$ in (5.4). From the Euler–Lagrangian equation for the minimiser $z_t = \tilde {z}^\varepsilon _t$ , we have

\begin{equation*} \frac {\,\mathrm{d}}{\,\mathrm{d} t} (2 B_\varepsilon (z_t)\dot {z}_t) = B_\varepsilon '(z_t)(\dot {z}_t)^2, \end{equation*}

leading to $B_\varepsilon '(z_t)\dot {z}_t^2+ 2 B_\varepsilon (z_t)\ddot {z}_t=0$ and thus

\begin{equation*} B_\varepsilon (z) \dot {z}^2=C_\varepsilon (x,y), \quad \text{for some constant $C_\varepsilon (x,y)$.} \end{equation*}

Upon solving this ODE for $z_t$ with the two boundary conditions $z(0)=x$ , $z(1)=y$ , we have

\begin{equation*} \sqrt {C_\varepsilon (x,y)}=\int _x^y \sqrt {B_\varepsilon (z)} \,\mathrm{d} z. \end{equation*}

Hence the infimum in (5.4) is given by

(5.11) \begin{equation} d^2_\varepsilon (x,y) = C_\varepsilon (x,y) = \left (\int _x^y \sqrt {B_\varepsilon (z)} \,\mathrm{d} z\right )^2. \end{equation}

As $B_\varepsilon (x)=B(\frac {x}{\varepsilon })$ , it is easy to verify that for any $x,y\in \Omega$ , there exist some integer $N_\varepsilon$ and $\delta \in (-1,1)$ , such that $y-x=N_\varepsilon \varepsilon + \delta \varepsilon$ and $N_\varepsilon \varepsilon \to |x-y|$ . Notice also $B(\cdot )$ is $1$ -periodic. Hence,

\begin{eqnarray*} d^2_\varepsilon (x,y) = \left ( \varepsilon \int _{\frac {x}{\varepsilon }}^{\frac {y}{\varepsilon }} \sqrt {B(s)} \,\mathrm{d} s\right )^2 &=& \left (\varepsilon N_\varepsilon \int _0^1 \sqrt {B(s)} \,\mathrm{d} s + \varepsilon \int _0^\delta \sqrt {B(s)} \,\mathrm{d} s\right )^2\\ &\stackrel {\longrightarrow }{\varepsilon \to 0} & |x-y|^2\left (\int _0^1 \sqrt {B(s)} \,\mathrm{d} s\right )^2 \,=\!:\, d^2_{\text{GH}}(x,y). \end{eqnarray*}

Notice that if one chooses $\mathcal R$ to be the identity map as the correspondence between the metric spaces $\mathcal{X}\,:\!=\,(\Omega , d_\varepsilon )$ and $\mathcal{Y}\,:\!=\, (\Omega , d_{\text{GH}})$ , then from (5.1), we have

\begin{equation*}D_{\text{GH}}(\mathcal{X},\mathcal{Y})\leq \frac {1}{2} \sup _{(x,x), (y,y)\in \mathcal{X}\times \mathcal{Y}}|d_\varepsilon (x,y)-d_{\text{GH}}(x,y)| \to 0.\end{equation*}

Hence the one dimensional metric space $(\Omega , d_\varepsilon )$ Gromov–Hausdorff converges to $(\Omega , d_{\text{GH}})$ . By [Reference Villani53, Theorem 28.6], the Wasserstein distance $W_\varepsilon$ defined in (5.2) also converges to the following limiting Wasserstein distance $W_{\text{GH}}$ in the Gromov–Hausdorff sense,

(5.12) \begin{equation} W_{\text{GH}}^2(\rho _0, \rho _1)\,:\!=\, \inf \left \{ \int \int d^2_{\text{GH}}(x,y) \,\mathrm{d} \gamma (x,y); \quad \int _{\Omega } \gamma ( x,\,\mathrm{d} y) = \rho _0(x) \,\mathrm{d} x, \,\, \int _{\Omega } \gamma (\,\mathrm{d} x, y) = \rho _1(y) \,\mathrm{d} y \right \}. \end{equation}

Again by [Reference Bernard and Buffoni6, Theorem AB], $W_{\text{GH}}$ can be equivalently written in the Benamou–Brenier formulation

(5.13) \begin{equation} W_{\text{GH}}^2(\rho _0, \rho _1)\,:\!=\, \inf \left \{ \int _0^1\int \rho _t(x)\langle \overline {C} v_t(x), v_t(x)\rangle \,\mathrm{d} x\,\mathrm{d} t,\quad (\rho _t, v_t)\in V(\rho _0, \rho _1) \right \} \end{equation}

with $\displaystyle \overline {C}=\left (\int _0^1 \sqrt {B(s)} \,\mathrm{d} s\right )^2.$

On the other hand, in one dimension, we can solve the cell problem (A.6) explicitly:

\begin{eqnarray*} \partial _y\big (D(x,y)\partial _y w(x,y)\big ) &=& -\partial _y\left (D(x,y)\right ), \quad \text{where}\,\,\,D(x,y) = \pi (x,y)B(y)^{-1},\\ \partial _y w(x,y) &=& -1 + \frac {C(x)}{D(x,y)} \quad \text{with}\,\,\,C(x) = \left (\int \frac {1}{D(x,y)}\,\mathrm{d} y\right )^{-1}. \end{eqnarray*}

Then (A.7) and (A.9) are given as

\begin{eqnarray*} \overline {D}(x) &=& \int D(x,y)\,\mathrm{d} y,\\ \overline {G}(x) &=& \int D(x,y) \left (-1 + \frac {C(x)}{D(x,y)}\right )\,\mathrm{d} y = -\int D(x,y)\,\mathrm{d} y + \left (\int \frac {1}{D(x,y)}\,\mathrm{d} y\right )^{-1},\\ \overline {B} &=& \left (\frac {\overline {D} + \overline {G}}{\overline {\pi }}\right )^{-1} = \overline {\pi }\int \frac {1}{D(x,y)}\,\mathrm{d} y = \overline {\pi }\int \frac {B(y)}{\pi (x,y)}\,\mathrm{d} y. \end{eqnarray*}

By the Cauchy–Schwarz inequality, we always have

\begin{align*} \overline {C} =\left (\int _0^1 \sqrt {B(s)} \,\mathrm{d} s\right )^2 =&\left (\int _0^1 \sqrt {\pi (x,y)}\sqrt {\frac {B(y)}{\pi (x,y)}} \,\mathrm{d} y\right )^2 \\ \leq & \left (\int \pi (x,y)\,\mathrm{d} y\right ) \left (\int \frac {B(y)}{\pi (x,y)}\,\mathrm{d} y\right ) = \overline {B}(x), \end{align*}

and the equality holds if and only if there exists some constant $c\gt 0$ such that

(5.14) \begin{eqnarray} \sqrt {\pi (x,y)} = c\sqrt {\frac {B(y)}{\pi (x,y)}}, \quad \text{i.e.}\quad \pi (x,y)=\pi (y) = c\sqrt {B(y)}. \end{eqnarray}

Hence, unless $\pi (y)=c\sqrt {B(y)}$ , we always have

\begin{equation*} d_{\text{GH}}(x,y) \lt \overline {d}(x,y)\quad \text{for all $x,y\in \Omega $} \end{equation*}

i.e. $W_{\text{GH}} \lt \overline {W}.$ As an afterthought, it seems not quite surprising that some condition, such as (5.14), is needed in order for $\overline {W}$ to be equal to $W_{\text{GH}}$ . We will elaborate upon this at the end of this section.

Next, we illustrate the $n$ -dimensional case by means of an example. From [Reference Braides12, Section 3.3], it is shown that the functional

(5.15) \begin{equation} {\mathcal F}_\varepsilon (z) = \int _0^1 \langle B_\varepsilon (z_t) \dot {z}_t, \dot {z}_t \rangle \,\mathrm{d} t, \quad \text{for} \quad z(\cdot )\in (H^1([0,1]))^n \quad \text{with}\quad z_0 = x, \quad z_1=y, \end{equation}

$\Gamma$ -converges with respect to the strong $L^2(0,1)$ -topology to

(5.16) \begin{equation} {\mathcal F}(z) = \int _0^1 \varphi (\dot {z}(t)) \,\mathrm{d} t \quad \text{for}\quad z(\cdot )\in (H^1([0,1]))^n, \quad \text{with}\quad z_0 = x, \quad z_1=y, \end{equation}

where the limiting integrand $\varphi$ is given by

(5.17) \begin{equation} \varphi (v) \,:\!=\, \lim _{T\to +\infty } \inf _{u\in (H^1_0([0,T]))^n} \left \{\frac {1}{T} \int _0^T \langle B(u(t)+vt) (\dot {u}(t)+v), \, \dot {u}(t)+v \rangle \,\mathrm{d} t\right \}. \end{equation}

Now following [Reference Braides12, Example 3.3], we consider $B_\varepsilon (z)=b(\frac {z}{\varepsilon })$ where $b$ is the following $1$ -periodic function on $[0,1]^n$ ,

\begin{equation*} b(y) = \left \{ \begin{array}{ll} \beta & \text{ if } y\in (0,1)^n;\\ \alpha & \text{ if for some }i,\, y_i\in \mathbb{Z}. \end{array} \right . \end{equation*}

If $n \alpha \lt \beta$ , one obtains that the limiting energy integrand $\varphi$ is given by

(5.18) \begin{equation} \varphi (v)=\alpha \left (\sum _{i=1}^n |v_i|\right )^2. \end{equation}

Using the property of $\Gamma$ -convergence [Reference Braides12],Theorem 1.21], we deduce also the convergence of the minimum value $d^2_\varepsilon$ of $\mathcal{F}_\varepsilon$ to the minimum value $d^2_{\text{GH}}$ of $\mathcal{F}$ , where

(5.19) \begin{equation} d_{\text{GH}}(x,y)= \sqrt {\alpha }\left (\sum _{i=1}^n |\hat {n}_i|\right )|y-x| = \sqrt {\alpha }\|y-x\|_{\ell ^1} \,\,\, \text{with $\hat {n}=\frac {y-x}{|y-x|}.$} \end{equation}

On the other hand, note that the value $\alpha$ is attained only on the $(n-1)$ -dimensional set $\bigcup _{i=1}^n\{y_i \in \mathbb Z\}$ . This set is invisible by $\overline {B}$ which is obtained by solving the elliptic cell problem (A.6). Hence the induced limiting Wasserstein distance $\overline {W}$ (5.5) with $\overline {d}$ defined in (5.7) is $\overline {d}(x,y)=\beta |x-y|$ for all $x,y\in \Omega$ . Thus, for this example, we have

\begin{equation*} d_{\text{GH}}(x,y)= \sqrt {\alpha } \|y-x\|_{\ell ^1} \leq \sqrt {\alpha n}\|y-x\|_{\ell ^2} \lt \sqrt {\beta } |y-x| = \overline {d}(x,y). \end{equation*}

Hence we have again $W_{\text{GH}} \lt \overline {W}$ .

We would like to point out that for the above example, the integrand $\varphi$ in (5.17) is always quadratic, or homogeneous of degree 2 in $p$ . (In fact, for any $\lambda \neq 0$ , by applying the change of variables $\tilde {t}=\lambda t, \tilde {u}(\tilde {t})=u(t)$ , it is easy to verify that $\varphi (\lambda v) = \lambda ^2 \varphi (v)$ .) However, the $\varphi$ in (5.18) is not bilinear in $p$ , in contrast to the $\varphi$ in (5.7):

\begin{equation*} \varphi (p) = \langle \bar {B} p, p \rangle . \end{equation*}

Below we give further remarks about the discrepancy between $\overline {W}$ and $W_{\text{GH}}$ .

1. (1) We first explain the condition (5.14). This is nothing but the fact that one can choose the Riemannian metric $(\mathbb{R}, g_\varepsilon )$ with $(g_\varepsilon )_{ij}(x)= B_\varepsilon (x),$ so that the Wasserstein distance on $(\mathbb{R}, g_\varepsilon )$ coincides with $W_\varepsilon$ . More precisely, the condition (5.14) implies the volume form on $(\mathbb{R}, g_\varepsilon )$ is

   (5.20) \begin{equation} \,\mathrm{d}\text{Vol} = \sqrt {|g_\varepsilon |}\,\mathrm{d} x = \sqrt {B_\varepsilon }\,\mathrm{d} x = c\pi _\varepsilon (x)\,\mathrm{d} x=c\pi (\frac {x}\varepsilon )\,\mathrm{d} x. \end{equation}
   Therefore, the heat flow on $(\mathbb{R}, g_\varepsilon )$ , in terms of the density function with respect to the volume element $\,\mathrm{d}\text{Vol}$ is given by
   (5.21) \begin{equation} \partial _t p_\varepsilon = \frac {1}{\sqrt {|g_\varepsilon |}}\nabla \cdot (\sqrt {|g_\varepsilon |}g^{ij}_\varepsilon \nabla p_\varepsilon )= \frac {1}{\pi _\varepsilon } \nabla \cdot (\pi _\varepsilon {B_\varepsilon ^{-1}} \nabla p_\varepsilon ). \end{equation}
   This equation, in terms of the density function $\rho _\varepsilon (x,t) = p_\varepsilon (x,t) \sqrt {|g_\varepsilon |} = p_\varepsilon (x,t) \pi _\varepsilon (x)$ , is exactly the $W_\varepsilon$ -gradient flow with respect to the relative entropy $E_\varepsilon$ in (2.1):
   (5.22) \begin{eqnarray} \partial _t \rho _\varepsilon = \nabla \cdot (\pi _\varepsilon {B_\varepsilon ^{-1}} \nabla \frac {\rho _\varepsilon }{\pi _\varepsilon })=\nabla \cdot \left ( \rho ^\varepsilon _t B_\varepsilon ^{-1} \nabla \frac {\delta E_\varepsilon }{\delta \rho }(\rho ^\varepsilon _t) \right ). \end{eqnarray}
   Therefore, condition (5.14) means that the discrepancy between $\overline {W}$ and $W_{\text{GH}}$ does not happen in one dimension when one considers homogenisation of heat flow on $(\mathbb{R}, g_\varepsilon )$ . In other words, the homogenised heat flow in one dimension naturally induces the same limiting distance as finding the limiting minimum path on $(\mathbb{R}, g_\varepsilon )$ . On the other hand, even in one dimension, the convergence of the discrete transport distance to continuous transport distance $W_2$ requires an isotropic mesh condition [Reference Gladbach, Kopfer, Maas and Portinale22],eq. (1.3)]. Without this condition, the discrete-to-continuous limiting distance in the Gromov–Hausdorff sense can be different from the continuous transport distance $W_2$ [Reference Gladbach, Kopfer, Maas and Portinale22], Theorem 1.1, Remarks 1.2 and 1.3].

2. (2) We believe that the above conclusion of $W_{\text{GH}} \lt \overline {W}$ is true in general, particularly in higher dimensions, even if we consider heat flow. This is because the Gromov–Hausdorff limit $d_{\text{GH}}$ of $d_\varepsilon$ involves finding the minimum or geodesic distance between two points as indicated in (5.4). This amounts to searching for the minimum path in the underlying spatial inhomogeneity. On the other hand, the $\overline {B}$ in the limiting induced distance $\overline {d}$ is found by solving an elliptic cell-problem (A.6) which requires taking some average of the spatial inhomogeneity. (Note that in contrast, in one dimension, any path will explore the whole inhomogeneous landscape.) Hence, in general $d_{\text{GH}}$ and $W_{\text{GH}}$ should be smaller than $\overline {d}$ and $\overline {W}$ . See also the discussion in [Reference Forkert, Maas and Portinale20, p. 4298] and the work [Reference Gladbach, Kopfer, Maas and Portinale22].