Introduction to the Metric Setting

In this article, we will introduce the gradient flows in the metric setting.

If one has a metric space $(X,d)$ and a l.s.c. function $F:X\to \mathbb{R}\cup\{+\infty\}$ (under suitable compactness assumptions to guarantee existence of the minimum), one can define
$$
x_{k+1}^{\tau}\in \arg\min_{x\in X}\left\{F(x)+\frac{d(x,x_k^{\tau})^2}{2\tau}\right\} \tag{9}
$$
and study the limit as $\tau\to 0$. Then we use the piecewise constant interpolation
$$
x^{\tau}(t):=x_k^{\tau}\qquad \text{for every } t\in ((k-1)\tau,k\tau].\tag{10}
$$
and study the limit of $x^{\tau}$ as $\tau\to 0$.

Definition 1. A curve $x:[0,T]\to X$ is called Generalized Minimizing Movements (GMM) if there exists a sequence of time steps $\tau_j\to 0$ such that the sequence of curves $x^{\tau_j}$, defined in (10) using the iterated solutions of (9), uniformly converges to $x$ in $[0,T]$.

We start from
$$
F(x_{k+1}^{\tau})+\frac{d(x_{k+1}^{\tau},x_k^{\tau})^2}{2\tau}\le F(x_k^{\tau})
$$
and
$$
\sum_{k=0}^{\ell}\frac{d(x_{k+1}^{\tau},x_k^{\tau})^2}{2\tau}\le F(x_0^{\tau})-F(x_{\ell+1}^{\tau})\le C\tau.
$$

The Cauchy–Schwarz inequality gives for $t<s$, $t\in [k\tau,(k+1)\tau]$ and $s\in [k’\tau,(k’+1)\tau]$ (which implies $|k’-k|\le \frac{|t-s|}{\tau}+1$),
$$
d(x^{\tau}(t),x^{\tau}(s))
\le d(x_k^{\tau},x_{k’}^{\tau})
\le \sum_{j=k}^{k’-1} d(x_{j+1}^{\tau},x_j^{\tau})
\le \left(\sum_{j=k}^{k’-1} d(x_{j+1}^{\tau},x_j^{\tau})^2\right)^{1/2}(k’-k)^{1/2}.
$$

For $t<s$, $t\in [i\tau,(i+1)\tau]$, $s\in [j\tau,(j+1)\tau]$ (which implies $j-i\le \frac{|t-s|}{\tau}+1$),

$$
d(x^{\tau}(t),x^{\tau}(s))
\le d(x_{i+1}^{\tau},x_{j+1}^{\tau})
\le \sum_{k=i+1}^{j} d(x_{k+1}^{\tau},x_k^{\tau})\le \left(\sum_{k=i+1}^{j} d(x_{k+1}^{\tau},x_k^{\tau})^2\right)^{1/2}(j-i)^{1/2}
\le \left(\sum_{k=0}^{\ell} d(x_{k+1}^{\tau},x_k^{\tau})^2\right)^{1/2}\left(\frac{|t-s|}{\tau}\right)^{1/2}
\le C|t-s|^{1/2}.
$$

This shows the curves $x^{\tau}$ are morally equi-Hölder with exponent $\frac12$. Since they start from the same $x^{\tau}(0)=x_0$, they are also equibounded. By applying Ascoli–Arzelà theorem, we can extract a converging subsequence.

Curves and geodesics in metric spaces

Definition 2. If $w:[0,1]\to X$ is a curve valued in the metric space $(X,d)$, we define the metric derivative of $w$ at the time $t$, denoted by $|w’|(t)$ through
$$
|w’|(t):=\lim_{h\to 0}\frac{d(w(t+h),w(t))}{|h|}
$$
provided this limit exists.

In the spirit of Rademacher theorem, it is possible to prove that if $w:[0,1]\to X$ is Lipschitz continuous, then the metric derivative $|w’|(t)$ exists for a.e. $t$. Moreover, we have for $t_0<t_1$
$$
d(w(t_0),w(t_1))\le \int_{t_0}^{t_1}|w’|(s)\ ds.
$$

Definition 3. A curve $w:[0,1]\to X$ is said to be absolutely continuous whenever there exists $g\in L^1[0,1]$ such that
$$
d(w(t_0),w(t_1))\le \int_{t_0}^{t_1} g(s)\ ds
\qquad \text{for every } t_0<t_1.
$$

The set of absolutely continuous curves defined on $[0,1]$ and valued in $X$ is denoted by $AC(X)$.

It is well-known that every absolutely continuous curve can be reparameterized in time (through a monotone increasing reparametrization) and become Lipschitz continuous, and the existence of the metric derivative is also true for $w\in AC(X)$ with i.e.t.

Definition 4. For a curve $w:[0,1]\to X$, let us define
$$
\operatorname{Length}(w):=\sup\left\{\sum_{k=0}^{n-1} d\bigl(w(t_k),w(t_{k+1})\bigr): n\ge 1,\ 0=t_0<t_1<\cdots<t_n=1\right\}.
$$

It is easy to see that all curves $w\in AC(X)$ satisfy
$$
\operatorname{Length}(w)\le \int_0^1 g(t)\ dt<+\infty.
$$

Also we can prove that for any curve $w\in AC(X)$, we have
$$
\operatorname{Length}(w)=\int_0^1 |w’|(t)\ dt.
$$

Definition 5. A curve $w:[0,1]\to X$ is said to be a geodesic between $x_0$ and $x_1\in X$ if $w(0)=x_0$, $w(1)=x_1$, and
$$
\operatorname{Length}(w)=\min\left\{\operatorname{Length}(\widetilde w): \widetilde w(0)=x_0,\ \widetilde w(1)=x_1\right\}.
$$

A space $(X,d)$ is said to be a length space if for every $x$ and $y$, we have
$$
d(x,y)=\inf\left\{\operatorname{Length}(w): w\in AC(X),\ w(0)=x,\ w(1)=y\right\}.
$$

A space $(X,d)$ is said to be a geodesic space if for every $x$ and $y$, we have
$$
d(x,y)=\min\left\{\operatorname{Length}(w): w\in AC(X),\ w(0)=x,\ w(1)=y\right\}.
$$
i.e. if it is a length space and there exist geodesics between arbitrary points.

In a length space, a curve $w:[t_0,t_1]\to X$ is said to be a constant-speed geodesic between $w(t_0)$ and $w(t_1)$ if it satisfies
$$
d(w(t),w(s))=\frac{|t-s|}{t_1-t_0}d(w(t_0),w(t_1))
\qquad \text{for all } t,s\in [t_0,t_1].
$$

The following three facts are equivalent:

1. $w$ is a constant-speed geodesic defined on $[t_0,t_1]$ and joining $x_0$ and $x_1$.
2. $w\in AC(X)$ and
   $$
   |w’|(t)=\frac{d(w(t_0),w(t_1))}{|t_1-t_0|}
   \qquad \text{a.e.}
   $$
3. $w$ solves
   $$
   \min\left\{\int_{t_0}^{t_1}|w’(t)|^p\ dt:\ w(t_0)=x_0,\ w(t_1)=x_1\right\}
   \qquad \text{for all } p>1.
   $$

Come back to the interpolation of the points obtained through the Minimizing Movement Scheme, if $(X,d)$ is a geodesic space, then the piecewise affine interpolation that we used in the Euclidean space may be helpfully replaced via a piecewise geodesic interpolation. This means defining a curve
$$
x^{\tau}:[0,T]\to X
$$
such that
$$
x^{\tau}(k\tau)=x_k^{\tau}
$$
and such that $x^{\tau}$ restricted to any interval $[k\tau,(k+1)\tau]$ is a constant-speed geodesic with speed equal to $\frac{d(x_k^{\tau},x_{k+1}^{\tau})}{\tau}$. Then the same equicontinuity will hold.

The next question is how to characterize the limit curve obtained when $\tau\to 0$. In a general metric space
$$
x’(t)=-\nabla F(x(t))
$$
has no meaning!

• EDE (Energy Dissipation Equality) viewpoint.

For $s<t$
$$
F(x(s))-F(x(t))
=\int_s^t -\nabla F(x(r))\cdot x’(r)\ dr
\le \int_s^t |\nabla F(x(r))|\ |x’(r)|\ dr
\le \int_s^t \left(\frac12|x’(r)|^2+\frac12|\nabla F(x(r))|^2\right)\ dr.
$$

Here the first inequality becomes equality if and only if $x’(r)$ and $\nabla F(x(r))$ are vectors with opposite direction for a.e. $r$, and the second inequality becomes equality if and only if their norms are the same. Therefore
$$
F(x(s))-F(x(t))
=\int_s^t \left(\frac12|x’(r)|^2+\frac12|\nabla F(x(r))|^2\right)\ dr
\qquad \text{for all } s<t
$$
if and only if
$$
x’(t)=-\nabla F(x(t))
\qquad \text{a.e. } t.
$$

• EVI (Evolution Variational Inequality) viewpoint.

If $F$ is $\lambda$-convex, the inequality that characterize the gradient is
$$
F(y)\ge F(x)+\frac{\lambda}{2}|y-x|^2+p\cdot (y-x)
\qquad \text{for all } y\in \mathbb{R}^n.
$$

Therefore
$$
\frac{d}{dt}\frac12|x(t)-y|^2=(y-x(t))\cdot \bigl(-x’(t)\bigr)
\le F(y)-F(x(t))-\frac{\lambda}{2}|x(t)-y|^2
\qquad \text{for all } y.
$$
which will be equivalent to
$$
-x’(t)\in \partial F(x(t)).
$$

By $\mathrm{EVI}_{\lambda}$, take two curves $x(t)$ and $y(s)$, we have
$$
\frac{d}{dt}\frac12 d(x(t),y(s))^2
\le F(y(s))-F(x(t))-\frac{\lambda}{2}d(x(t),y(s))^2
\tag{12}
$$
and
$$
\frac{d}{ds}\frac12 d(x(t),y(s))^2
\le F(x(t))-F(y(s))-\frac{\lambda}{2}d(x(t),y(s))^2.
\tag{13}
$$

Define
$$
E(t)=\frac12 d(x(t),y(t))^2,
\qquad
G(t,s)=\frac12 d(x(t),y(s))^2.
$$
Then
$$
\frac{d}{dt}E(t)=\frac{d}{dt}G(t,t)=\partial_t G(t,t)+\partial_s G(t,t)
\le -\lambda d(x(t),y(t))^2=-2\lambda E(t).
$$

By Gronwall inequality, this provides uniqueness and stability.

Reference

Santambrogio, F. {Euclidean, metric, and Wasserstein} gradient flows: an overview. Bull. Math. Sci. 7, 87–154 (2017).

The cover image in this article was taken at Château de Chillon in Switzerland.