1 Introduction↩︎

The minimizing movement scheme, introduced by De Giorgi in [1] and further developed by Ambrosio in [2], is one of the basic variational methods for constructing evolutions from time-discrete minimization problems. Its metric formulation has become a central tool in the theory of gradient flows, especially after the development of the general theory in metric spaces and in Wasserstein spaces; we refer to [3]–[6] for the general background. A fundamental example is the Jordan–Kinderlehrer–Otto scheme for the Fokker–Planck equation [7], where the evolution is obtained by iteratively minimizing an energy penalized by the squared \(W_2\)-distance from the previous time step (see also [8]).

The classical Wasserstein theory is particularly effective for displacement-convex energies, such as internal, potential, and interaction energies (see e.g. Chapter 9 of [3]). Several natural functionals in the calculus of variations, however, have a different form. This is the case for Sobolev-type and isoperimetric ratios, where a first-order quantity is normalized by a Lebesgue norm. Typical examples are \[\begin{align} \label{I951} \mathcal{S}_r(f):= \frac{\lVert\nabla f\rVert_{L^r}}{\lVert f\rVert_{L^{r^*}}} \quad \text{and}\quad \mathrm{Isop}(f):= \frac{\lVert\delta f\rVert}{\lVert f\rVert_{L^{n/(n-1)}}}, \end{align}\tag{1}\] where \(r^*:=nr/(n-r)\) is the Sobolev conjugate of \(1<r<n\) and \(\lVert\delta f\rVert\) is the total variation of \(f\). These are the scale-invariant quantities associated with the sharp Sobolev and isoperimetric inequalities. In particular, both functionals admit global minimizers, up to the natural invariances of the problem. The global minimizers of the Sobolev quotient \(\mathcal{S}_r\) are the Aubin–Talenti functions, as shown in the classical works of Aubin and Talenti [9], [10]. On the other hand, the global minimizers of the isoperimetric quotient \(\text{\normalfont Isop}\) are given by multiples of characteristic functions of balls, in accordance with the sharp isoperimetric inequality for sets of finite perimeter; see the works of De Giorgi and Fleming–Rishel [11], [12]. Their connection with optimal transport is also well established: mass-transportation methods give proofs of sharp Sobolev and Gagliardo–Nirenberg inequalities [13], quantitative anisotropic isoperimetric inequalities [14], and sharp stability results for anisotropic Sobolev-type inequalities in \(BV\) [15].

The interaction between first-order variational quantities and optimal transport has already appeared in several works. The Wasserstein gradient flow of the total variation and the corresponding TV–JKO scheme were studied by Carlier–Poon in [16] and more recently by Lin–Santambrogio in [17], and related Euler–Lagrange equations were later considered in the work by Chambolle–Duval–Machado [18].

The aim of this paper is to introduce a metric framework in which transportation of mass and regularity of the density are both part of the topology. For \((p,q)\in [1,\infty]\times (1,\infty]\), we consider the class \[\begin{align} \text{\normalfont PL}_q^p({{\mathbf{R}}^n}) := \left\{\mu=f{\mathscr{L}^n}:\mu\in\mathcal{P}_q({{\mathbf{R}}^n}),\,f\in L^p({{\mathbf{R}}^n}) \right\}, \end{align}\] endowed with the metric \[\begin{align} {\mathfrak{d}}_q^p(f{\mathscr{L}^n},g{\mathscr{L}^n}):= W_q(f{\mathscr{L}^n},g{\mathscr{L}^n})+\lVert f-g\rVert_{L^p({{\mathbf{R}}^n})}. \end{align}\] The Wasserstein term in \({\mathfrak{d}}_q^p\) controls the displacement of mass, while the \(L^p\) term controls the density in a topology which is natural for Sobolev and isoperimetric quantities. This makes \(\text{\normalfont PL}_q^p(\mathbf{R}^n)\) a suitable setting for variational problems whose value depends both on transport and on analytic properties of the density.

One of the main results of the paper is the Eulerian characterization of absolutely continuous curves in \(\text{\normalfont PL}_q^p(\mathbf{R}^n)\) given by Theorem 13, for \(1<p\leq \infty\), and by Theorem 14, for the degenerate case \(p=1\). Roughly speaking, we prove that a curve \(\mu_{(\cdot)}:[0,T]\to \text{\normalfont PL}_q^p({{\mathbf{R}}^n})\) is absolutely continuous with respect to \({\mathfrak{d}}_q^p\) if and only if it admits a Borel velocity field \((t,x)\mapsto v_t(x)\) solving the continuity equation \[\begin{align} \partial_t\mu_t+\text{\normalfont div}(v_t\mu_t)=0 \end{align}\] and satisfying the first-order integrability condition \[\begin{align} \int_0^T \left( \lVert v_t\rVert_{L^q(\mu_t)} +\lVert\text{\normalfont div}(v_t\mu_t)\rVert_{L^p({{\mathbf{R}}^n})} \right)\,dt <\infty . \end{align}\] Moreover, the metric derivative with respect to \({\mathfrak{d}}_q^p\) is bounded from above by the sum \(\lVert v_t\rVert_{L^q(\mu_t)} + \lVert\text{\normalfont div}(v_t\mu_t)\rVert_{L^p}\).

The endpoint space \((\text{\normalfont PL}^\infty_\infty({{\mathbf{R}}^n}),{\mathfrak{d}}_\infty^\infty)\) emerges from the study of the minimizing movements of the isoperimetric functional. Indeed, the natural domain for the isoperimetric ratio is the family \(\mathcal{F}\) of bounded Borel subsets with positive measure and it is defined on \(\mathcal{F}\) as \[\begin{align} \frac{\text{Per}(\Omega)}{({\mathscr{L}^n}(\Omega))^{(n-1)/n}}, \end{align}\] where \(\text{Per}(\Omega)\) denotes the perimeter of the set \(\Omega\in \mathcal{F}\). This family is embedded in the complete metric space \((\mathcal{P}_\infty({{\mathbf{R}}^n}),W_\infty)\) via the map \(\iota: \mathcal{F}\hookrightarrow \mathcal{P}_\infty({{\mathbf{R}}^n})\), \(\iota(\Omega) := ({\mathscr{L}^n}(\Omega))^{-1}{\mathscr{L}^n}{\mathrel{\makebox[7pt][c]{\rule{0.4pt}{6.75pt}\rule{5.5pt}{.4pt}}}}\Omega\), thereby endowing it with a canonical notion of transportation cost, and the isoperimetric ratio of \(\iota(\Omega)\) is still well-defined as \(\text{\normalfont Isop}(f_\Omega)\), with \(f_\Omega= ({\mathscr{L}^n}(\Omega))^{-1}\chi_\Omega\) being the density of the probability \(\iota(\Omega)\) and \(\text{\normalfont Isop}\) as in 1 . However, the subfamily \(\iota(\mathcal{F})\) does not form a complete metric subspace of the \(\infty\)-Wasserstein space. Moreover, \(\iota(\mathcal{F})\) is not convex and is not stable under the relaxed limits naturally arising in variational compactness arguments. To resolve these issues, we convexify \(\iota(\mathcal{F})\) into the set \(\text{\normalfont PL}_\infty^\infty({{\mathbf{R}}^n})\) of probabilities with density in \(L^\infty\) and bounded support, and incorporate the \(L^\infty\)-norm into the metric. The resulting metric space is complete, convex, has a canonical notion of transportation cost and the isoperimetric ratio of its elements is well-defined.

Another main result of the paper concerns the existence of absolutely continuous generalized minimizing movements for the isoperimetric ratio. In the space \(\text{\normalfont PL}_\infty^\infty(\mathbf{R}^n)\), we study the implicit scheme \[\begin{align} \mu_{k+1}^{\tau}\in \operatorname*{argmin}_{\substack{\mu\in \text{\normalfont PL}_\infty^\infty({{\mathbf{R}}^n})\\ \mu=f{\mathscr{L}^n}}} \left\{\text{\normalfont Isop}(f)+\frac{1}{2\tau}\left({\mathfrak{d}}_\infty^\infty(\mu,\mu_k^\tau)\right)^2\right\}, \quad \tau>0. \end{align}\] We prove that this scheme admits absolutely continuous generalized minimizing movements (GMMs). The proof is based on the abstract theory of minimizing-movement presented in Chapter 2 of the book by Ambrosio–Gigli–Savaré [3], together with compactness and lower semicontinuity properties provided by the metric \({\mathfrak{d}}_\infty^\infty\).

The \(\infty\)-Wasserstein space \((\mathcal{P}_\infty({{\mathbf{R}}^n}),W_\infty)\) plays a central role in the paper. Since \({\mathfrak{d}}_\infty^\infty\) contains the uniform transportation distance \(W_\infty\), the study of absolutely continuous curves in \(\text{\normalfont PL}_\infty^\infty(\mathbf{R}^n)\) requires an analogue of the classical continuity-equation characterization of Wasserstein absolutely continuous curves. For \(1<q<\infty\), this characterization is classical and follows from the dynamical theory of Wasserstein spaces; see, for instance, [3], [5], [19]–[21]. For \(q=\infty\), the corresponding statement can be regarded as the formal limit as \(q\to\infty\) of this theory and is closely related to the literature on dynamical transport distances and on the \(W_\infty\) distance [22]–[24]. We give a self-contained proof of the form needed here. In particular, in the appendix we show the Benamou–Brenier formula \[\begin{align} W_\infty(\mu,\nu) = \min\left\{\lVert v\rVert_{L^\infty(\tilde{\mu})}: \begin{matrix} \tilde{\mu} :={\mathscr{L}}^1{\mathrel{\makebox[7pt][c]{\rule{0.4pt}{6.75pt}\rule{5.5pt}{.4pt}}}}[0,1] \otimes(\mu_t)_t\hfill\\ \partial_t\mu_t + \text{\normalfont div}(v_t\mu_t) = 0\text{ on }[0,1]\hfill\\ \mu_{(\cdot)}\text{ narrowly continuous}\hfill\\ \mu_0 = \mu\text{ and } \mu_1=\nu\hfill\\ \end{matrix} \right\} \end{align}\] (see [25] for the original statement with \(q=2\) and [20] for the analogous version with \(1<q<\infty\)), as well as an action-minimization formula \[\begin{align} W_\infty(\mu,\nu) = \min\left\{\lVert\mathscr{A}_\infty\rVert_{L^\infty(\eta)}: \begin{matrix} \eta\in \mathcal{P}({{\mathscr{C}}}^0([0,1];\mathscr{S})),\hfill \\ (e_0)_\#\eta = \mu,\hfill \\(e_1)_\#\eta =\nu\hfill \end{matrix}\right\}, \end{align}\] where \(\mathscr{A}_\infty(\omega) = \lVert\,|\dot{\omega}|\,\rVert_{L^\infty([0,1])}\) is the \(\infty\)-action and \((\mathscr{S},\texttt{d})\) is a general Polish geodesic space.

The paper is organized as follows. In Section 2 we collect the notation and the preliminary material on measure theory, optimal transport, absolute continuity in metric and Banach spaces, and GMMs. In Section 3 we introduce the spaces \((\text{\normalfont PL}_q^p({{\mathbf{R}}^n}),{\mathfrak{d}}_p^q)\), prove their basic metric properties, and establish the existence of GMMs for the isoperimetric ratio in \(\text{\normalfont PL}_\infty^\infty({{\mathbf{R}}^n})\). Section 4 is devoted to absolutely continuous curves in \(\mathcal{P}_\infty({{\mathbf{R}}^n})\). In Section 5 we prove the main characterization theorems for absolutely continuous curves in \(\text{\normalfont PL}_q^p({{\mathbf{R}}^n})\), treating separately the cases \(1<p\leq\infty\) and \(p=1\). Section 6 contains final remarks and open problems. The appendix contains the dynamical formulations for \(W_\infty\) as well as a superposition principle.

Acknowledgments. I would like to thank my supervisor, Prof. Dr. Zoltán Balogh, for the many fruitful discussions we have had during the development of this work.

2 Notation and preliminaries↩︎

2.1 Measure theory.↩︎

Let \(({{\mathscr{S}}},{\normalfont\texttt{d}})\) be a metric space. The Borel \(\sigma\)-algebra of \({{\mathscr{S}}}\) is the \(\sigma\)-algebra \(\mathcal{B}({{\mathscr{S}}})\) generated by the open subsets of \({{\mathscr{S}}}\). For every integer \(m\geq 1\), denote by \(\mathcal{M}({{\mathscr{S}}};\mathbf{R}^m)\) (resp. \(\mathcal{M}({{\mathscr{S}}})\), if \(m=1\)) the family of finite \(\mathbf{R}^m\)-valued (resp. real-valued) Borel measures on \({{\mathscr{S}}}\). The family of Borel non-negative measures is the subset \(\mathcal{M}_+({{\mathscr{S}}})\subseteq \mathcal{M}({{\mathscr{S}}})\) containing only the measures \(\mu\) such that \(\mu(B)\geq 0\) for every Borel subset \(B\in \mathcal{B}({{\mathscr{S}}})\), and the family of Borel probability measures is the subset \(\mathcal{P}({{\mathscr{S}}})\subseteq \mathcal{M}_+({{\mathscr{S}}})\) containing only the non-negative measures \(\mu\) such that \(\mu({{\mathscr{S}}})=1\).

The space of \(\mathbf{R}^m\)-valued measures is a vector space, and forms a Banach space when endowed with the total variation norm \(\lVert\cdot\rVert_{\mathrm{\small tv}}\) defined by \[\begin{align} \lVert E\rVert_{\mathrm{\small tv}}:=\sup\left\{\int_{{\mathscr{S}}}\langle v,dE\rangle : v\in {{\mathscr{C}}}^0({{\mathscr{S}}};\mathbf{R}^m),\,\lVert v\rVert_{{{\mathscr{C}}}^0({{\mathscr{S}}};\mathbf{R}^m)}\leq 1\right\}. \end{align}\] Moreover, by Riesz’ representation theorem, the space \((\mathcal{M}({{\mathscr{S}}};\mathbf{R}^m),\lVert\cdot\rVert_{\mathrm{\small tv}})\) is isometrically isomorphic to the topological dual \(({{\mathscr{C}}}^0_0({{\mathscr{S}}};\mathbf{R}^m),\lVert\cdot\rVert_{{{\mathscr{C}}}^0({{\mathscr{S}}};\mathbf{R}^m)})^*\), where \({{\mathscr{C}}}^0_0({{\mathscr{S}}};\mathbf{R}^m)\) (or simply \({{\mathscr{C}}}^0_0({{\mathscr{S}}})\), if \(m=1\)) denotes the space of \(\mathbf{R}^m\)-valued continuous functions that vanish at \(\infty\).

We say that a sequence of probability measures \((\mu_j)_{j\geq 1}\subseteq \mathcal{P}({{\mathscr{S}}})\) converges narrowly to \(\mu\in \mathcal{M}_+({{\mathscr{S}}})\), and write \(\mu_j\xrightarrow{\text{narrow}}\mu\) if \[\begin{align} \lim_{j\to\infty}\int_{{\mathscr{S}}}\psi\,d\mu_j = \int_{{\mathscr{S}}}\psi\,d\mu \quad \forall \psi\in {{\mathscr{C}}}^0_b({{\mathscr{S}}}). \end{align}\]

Let \(F:{{\mathscr{S}}}\to \mathscr{T}\) be a Borel map between two metric spaces \({{\mathscr{S}}}\) and \(\mathscr{T}\), and let \(\xi\in \mathcal{M}_+({{\mathscr{S}}})\). The push-forward of \(\xi\) through \(F\) is the measure \(F_\#\xi\in \mathcal{M}_+(\mathscr{T})\) defined by \[\begin{align} F_\#\xi(B) := \xi(F^{-1}(B))\quad \forall B\in \mathcal{B}(\mathscr{T}), \end{align}\] or equivalently \[\begin{align} \int_{\mathscr{T}}\varphi\,dF_\#\xi = \int_{{\mathscr{S}}}\varphi\circ F\,d\xi \quad \forall \varphi:\mathscr{T}\to [0,\infty] \text{ Borel measurable}. \end{align}\]

2.2 Absolute continuity in metric and dual Banach spaces.↩︎

Fix a Banach space \((X,\lVert\cdot\rVert_X)\). We say that a curve \(\omega:[0,T]\to X\) is differentiable almost everywhere if there exists a function \(\dot{\omega}:[0,T]\to X\), called derivative of \(\omega\) such that \[\begin{align} \lim_{{h\to 0}} \lVert\frac{\omega(t+h)-\omega(t)}{h} - \dot{\omega}(t)\rVert_X=0 \quad \text{for almost every }0\leq t\leq T. \end{align}\]

Let \(X\) be a separable Banach space. A function \(\omega:[0,T]\to X\) is Bochner measurable if \(\omega\) is Borel measurable with respect to the topology induced by the norm \(\lVert\cdot\rVert_X\). The \(r\)-Bochner space \(L^r([0,T];X)\) is then the space of Bochner measurable functions \(\omega:[0,T]\to X\) such that the function \(t\mapsto \lVert\omega(t)\rVert_X\) belongs to \(L^r([0,T])\).

The Bochner integral of a simple function \(\sigma:[0,T]\to X\) defined by \(\sigma(t):= \sum_{k=1}^N x_k \chi_{A_k}(t)\), with \(x_1,\dots,x_N\in X\) and \(A_1,\dots,A_N\in \mathcal{B}([0,T])\) pairwise disjoint, is defined as the sum \(\sum_{k=1}^N x_k{\mathscr{L}}^1(A_k)\in X\). For general curves \(\omega\in L^p([0,T];X)\), the Bochner integral is defined as the limit \[\begin{align} \int_0^T\omega(t)\,dt := \lim_{j\to\infty}\sum_{k=1}^{N_j} x_{k,j}{\mathscr{L}}^1(A_{k,j})\in X, \end{align}\] where \(\sigma_j:= \sum_{k=1}^{N_j} x_{k,j} \chi_{A_{k,j}}\) is any sequence of simple functions such that \[\begin{align} \lim_{j\to\infty}\int_X\lVert\omega(t)-\sigma_j(t)\rVert_X\,dt = 0. \end{align}\]

A Banach space \((X,\lVert\cdot\rVert_{X})\) is called dual Banach space if there exists a Banach space \((Y,\lVert\cdot\rVert_Y)\) such that \((X,\lVert\cdot\rVert_{X})\) is isometrically isomorphic to the dual \((Y,\lVert\cdot\rVert_Y)^*\); in this case, \(Y\) is called predual space of \(X\). A Banach space \(X\) is called reflexive if \(X\) is isometrically isomorphic to its bi-dual \(((X,\lVert\cdot\rVert_X)^*)^*\).

If \(X\) is a dual Banach space, \(Y\) is a predual of \(X\) and \(\langle\cdot,\cdot\rangle:X\times Y\to \mathbf{R}\) is the duality pairing, then the curve \(\omega:[0,T]\to X\) is said to be weakly\(^*\) differentiable almost everywhere if there exists a function \(\dot{\omega}:[0,T]\to X\), called weak\(^*\) derivative of \(\omega\), such that \[\begin{align} \lim_{h\to 0} \langle \frac{\omega(t+h)-\omega(t)}{h}, y\rangle = \langle \dot{\omega}(t),y\rangle\quad \forall y\in Y\quad\text{for almost every }0\leq t\leq T. \end{align}\]

We say that \(\omega:[0,T]\to X\) is weakly\(^*\) measurable if the real-valued map \(t\mapsto \langle \omega(t),y\rangle\) is Borel measurable for every \(y\in Y\). The weak\(^*\)-\(L^r\) space \(L^r_{w^*}([0,T];X)\) is the space of weakly\(^*\) measurable functions \(\omega:[0,T]\to X\) such that the function \(t\mapsto \lVert\omega(t)\rVert_X\) belongs to \(L^r([0,T])\). The weak\(^*\) integral of a function \(\omega\in L^r_{w^*}([0,T];X)\) is an element of \(X\) denoted by \(\int_0^T\omega(t) dt\) with the property \[\begin{align} \langle \int_0^T\omega(t) dt,y\rangle = \int_0^T \langle \omega(t),y\rangle \,dt\quad \forall y\in Y. \end{align}\]

We shall recall the following result (we refer to Remark 2.2 of [21], see also [2], [26])

2.3 Minimizing movements.↩︎

We refer to Chapter 2 of [3] for all the results presented in this subsection.

Consider the following assumptions:

1. \(({{\mathscr{S}}},{\normalfont\texttt{d}})\) is a complete metric space;

2. \(\sigma\) is a Hausdorff topology on \({{\mathscr{S}}}\), called weak topology, that is compatible with the metric topology, i.e.

   1. \(\sigma\) is weaker than the metric topology, and

   2. \({\normalfont\texttt{d}}:{{\mathscr{S}}}\times{{\mathscr{S}}}\to[0,\infty)\) is \(\sigma\)-sequentially lower semicontinuous;

3. \(\varphi:{{\mathscr{S}}}\to [0,\infty]\) is a functional such that:

   1. \(\varphi\) is \(\sigma\)-sequentially lower semicontinuous, and

   2. if \((x_j)_{j\geq 1}\subseteq \{\varphi\leq \lambda\}\) is a \({\normalfont\texttt{d}}\)-bounded sequence, then exists a subsequence \((x_{j_k})\preceq(x_j)\) and \(x_*\in\mathscr{S}\) such that \(x_{j_k}\xrightarrow{\sigma}x_*\).

Let \(\varphi:{{\mathscr{S}}}\to [0,\infty]\) be a functional as in [topAss3] and define \(\Phi:{{\mathscr{S}}}\times (0,\infty)\times {{\mathscr{S}}}\to [0,\infty]\) \[\begin{align} \Phi(x;\tau,\overline{x}):= \varphi(x) + \frac{1}{2\tau}{\normalfont\texttt{d}}^2(x,\overline{x}). \end{align}\] For any \(\tau\ge0\), the \(\tau\)-resolvent operator of \(\varphi\) is the set-valued functional \(J_\tau[\cdot]:{{\mathscr{S}}}\to 2^{{{\mathscr{S}}}}\) \[\begin{align} J_\tau[x]:= \operatorname*{argmin}_{{\mathscr{S}}}\,\Phi(\cdot;\tau,x):=\left\{y \in {{\mathscr{S}}}: \Phi(y;\tau,x)= \inf_{z\in {{\mathscr{S}}}} \Phi(z;\tau,x)\right\}\subseteq {{\mathscr{S}}}. \end{align}\]

A partition of steps is a family \({\overline{\tau}}:=\{\tau_j\}_{j\geq 1}\subseteq (0,\infty)\) such that:

1. \(|{\overline{\tau}}|:=\sup_{j\geq 1}\tau_j\le\infty\);

2. \(\sum_{j\geq 1}\tau_j = \infty\).

Given a partition of steps \({\overline{\tau}}\), the corresponding partition of times is the collection \(\{t_j^{\overline{\tau}}\}_{j\geq 0}\subseteq[0,\infty)\) of the numbers defined by \[\begin{align} t_0^{\overline{\tau}}:=0,\quad t^{\overline{\tau}}_j:=\sum_{k=1}^j\tau_k \quad \forall j\geq 1. \end{align}\] For every \(j\geq 1\), the \(j\)-th interval associated with the partition of steps \({\overline{\tau}}\) is the interval \(I^{\overline{\tau}}_j:=(t^{\overline{\tau}}_{j-1},t^{\overline{\tau}}_j]\).

2.4 Wasserstein spaces.↩︎

Let \(n\geq 1\) be a fixed dimension. We denote by \(({{\mathbf{R}}^n},\text{\normalfont d}_{{\mathbf{R}}^n})\) the Euclidean \(n\)-dimensional metric space. Consider the projections \(\pi_j:{{\mathbf{R}}^n}\times{{\mathbf{R}}^n}\to {{\mathbf{R}}^n}\) defined by \(\pi_j(x_1,x_2):=x_j\) for \(j\in \{1,2\}\). Given two probability measures \(\mu,\nu\in \mathcal{P}({{\mathbf{R}}^n})\) we define the set of couplings of \(\mu\) and \(\nu\) as \[\begin{align} \Gamma(\mu,\nu):=\left\{\gamma\in \mathcal{P}({{\mathbf{R}}^n}\times {{\mathbf{R}}^n}): (\pi_1)_\#\gamma = \mu,\, (\pi_2)_\#\gamma = \nu\right\}. \end{align}\]

It turns out ([3], [4], [20], and [23], [24] for the case \(q=\infty\)) that for every \(1\leq q\leq \infty\) \((\mathcal{P}_q({{\mathbf{R}}^n}),W_q)\) is a complete metric space and the infimum in \(W_q(\mu,\nu)\) is attained for every \(\mu,\nu\in \mathcal{P}_q({{\mathbf{R}}^n})\). In the sequel, we will denote by \(\Gamma_q(\mu,\nu)\) the set of couplings in \(\gamma\in \Gamma(\mu,\nu)\) such that \[\begin{align} W_q(\mu,\nu) =\lVert\text{\normalfont d}_{{\mathbf{R}}^n}(\cdot,\cdot)\rVert_{L^q(\gamma)}, \end{align}\] for every \(1\leq q \leq\infty\). The elements of \(\Gamma_q(\mu,\nu)\) will be also called \(q\)-optimal couplings of \(\mu\) and \(\nu\).

The topology induced by \(W_q\) is in general stronger than the one of the narrow convergence of measures, namely \(W_q\)-convergence always implies narrow convergence of measure. If \(1\leq q <\infty\) and \((\mu_j)_{j\geq 1}\subseteq \mathcal{P}_q({{\mathbf{R}}^n})\) is equi-compactly supported (i.e. there exists \(K\subseteq {{\mathbf{R}}^n}\) compact such that \(\text{\normalfont spt}\,\mu_j\subseteq K\) for all \(j\geq 1\)), then narrow convergence and \(W_q\)-convergence coincide. However, there exist equi-compactly supported sequences \((\mu_j)_{j\geq 1}\subseteq \mathcal{P}_\infty({{\mathbf{R}}^n})\) that converge narrowly but don’t admit \(W_\infty\)-limit (an easy example is the sequence \(\mu_j:= (1-j^{-1})\delta_0 + j^{-1}\delta_1\) in \(\mathbf{R}\)).

A useful characterization of the \(W_\infty\)-metric is the following: \[\begin{align} \label{eq95Winfty} W_\infty(\mu,\nu)= \inf\{\varepsilon\ge0 : \mu(A)\leq \nu(A_\varepsilon)\,\,\forall A\in \mathcal{B}({{\mathbf{R}}^n})\}, \end{align}\tag{3}\] where \(A_\varepsilon:= \{y\in {{\mathbf{R}}^n}: \text{\normalfont d}_{{\mathbf{R}}^n}(y,A)\le\varepsilon\}\) is the \(\varepsilon\)-neighborhood of \(A\).

2.5 Absolute continuity in \(\mathcal{P}_q({{\mathbf{R}}^n})\) for \(1<q<\infty\).↩︎

3 The \(\text{\normalfont PL}_q^p\)-spaces and applications↩︎

Let \((\mu_j)_{j\geq 1}\subseteq \text{\normalfont PL}^p_q({{\mathbf{R}}^n})\) be a \({\mathfrak{d}}_q^p\)-Cauchy sequence. Then \((\mu_j)_{j\geq 1}\subseteq \mathcal{P}_q({{\mathbf{R}}^n})\) is \(W_q\)-Cauchy and \((d\mu_j/d{\mathscr{L}^n})_{j\geq 1}\subseteq L^p({{\mathbf{R}}^n})\) is \(L^p\)-Cauchy. By completeness of \((\mathcal{P}_q({{\mathbf{R}}^n}),W_q)\) and of \((L^p({{\mathbf{R}}^n}),\lVert\cdot\rVert_{L^p})\) respectively, there exist \(\mu\in \mathcal{P}_q({{\mathbf{R}}^n})\) and \(f\in L^p({{\mathbf{R}}^n})\) such that \[\begin{align} \lim_{j\to\infty} W_q (\mu_j,\mu) = 0 \quad \text{and}\quad \lim_{j\to\infty}\lVert\frac{d\mu_j}{d{\mathscr{L}^n}}-f\rVert_{L^p}=0. \end{align}\] As \(W_q\)-convergence implies weak\(^*\)-convergence of measures, for every test function \(\psi \in {{\mathscr{C}}}^\infty_c({{\mathbf{R}}^n})\) we have \[\begin{align} \int_{{\mathbf{R}}^n}\psi\,d\mu = \lim_{j\to\infty}\int_{{\mathbf{R}}^n}\psi \,d\mu_j = \lim_{j\to \infty}\int_{{\mathbf{R}}^n}\psi\frac{d\mu_j}{d{\mathscr{L}^n}}\,d{\mathscr{L}^n}= \int_{{\mathbf{R}}^n}\psi\,f\,d{\mathscr{L}^n}. \end{align}\] Therefore \(\mu = f{\mathscr{L}^n}\in \text{\normalfont PL}^p_q({{\mathbf{R}}^n})\) and \({\mathfrak{d}_q^p}(\mu_j,\mu)\to 0\) as \(j\to \infty\).

We now present the main motivation for the development of the theory of \(\text{\normalfont PL}_q^p\) space. Consider the isoperimetric functional \(\text{\normalfont Isop}:\text{\normalfont PL}_\infty^\infty({{\mathbf{R}}^n})\to [0,\infty]\) defined by \[\begin{align} \label{def95Isop} \text{\normalfont Isop}(f{\mathscr{L}^n}):= \frac{\lVert\delta f\rVert}{\lVert f\rVert_{L^{n/(n-1)}}}, \end{align}\tag{6}\] where \[\begin{align} \lVert\delta f\rVert:=\sup\left\{\int_{{\mathbf{R}}^n}f\text{\normalfont div}\,\Psi\,d{\mathscr{L}^n}: \Psi\in {{\mathscr{C}}}^1_c({{\mathbf{R}}^n};{{\mathbf{R}}^n}),\,\lVert\Psi\rVert_{{{\mathscr{C}}}^0}\leq 1\right\} \end{align}\] is the total variation of \(f\). Observe that for every \(f{\mathscr{L}^n}\in {\text{\normalfont PL}}_\infty^\infty({{\mathbf{R}}^n})\), \(f\in L^p({{\mathbf{R}}^n})\) for every \(1\leq p\leq \infty\). In particular \(f\in L^{n/(n-1)}({{\mathbf{R}}^n})\) and therefore the denominator of \(\text{\normalfont Isop}(f{\mathscr{L}^n})\) is always finite.

Finally, the weak topology \(\sigma\) that we consider is the topology of \(L^{\frac{n}{n-1}}\)-convergence of densities, namely the topology defined by the following condition \[\begin{align} \mu_j\xrightarrow{\sigma}\mu :\iff \lim_{j\to\infty}\lVert\frac{d\mu_j}{d{\mathscr{L}^n}}-\frac{d\mu}{d{\mathscr{L}^n}}\rVert_{L^{n/(n-1)}} =0. \end{align}\]

If \((\mu_j)_{j\geq 1}\subseteq {\text{\normalfont PL}}_\infty^\infty({{\mathbf{R}}^n})\) is a \(\mathfrak{d}_\infty^\infty\)-converging sequence, and \(\mu\) is its \(\mathfrak{d}_\infty^\infty\)-limit, then 3 implies that \((\mu_j)_{j\geq 1}\) is equi-compactly supported. By standard interpolation of Lebesgue spaces, it follows that \((d\mu_j/d{\mathscr{L}^n})_{j\geq 1}\) converges to \(d\mu/d{\mathscr{L}^n}\) in \(L^{\frac{n}{n-1}}\). Thus \(\mu_j\xrightarrow{\sigma}\mu\). This proves the first statement.

Let \(\mu_j,\nu_j,\mu,\nu\in {\text{\normalfont PL}}_\infty^\infty({{\mathbf{R}}^n})\) for all \(j\geq 1\) and suppose \(\mu_j\xrightarrow{\sigma}\mu\) and \(\nu_j\xrightarrow{\sigma}\nu\). We need to show \[\begin{align} \label{eq95assTop2951} \mathfrak{d}_\infty^\infty(\mu,\nu)\leq \liminf_{j\to \infty} \mathfrak{d}_\infty^\infty(\mu_j,\nu_j). \end{align}\tag{7}\] Without loss of generality, up to the extraction of a subsequence we may suppose the limit to be attained and finite, and, up to the extraction of a further subsequence, we may also suppose that \[\begin{align} \left(\frac{d\mu_j}{d{\mathscr{L}^n}}(x),\frac{d\nu_j}{d{\mathscr{L}^n}}(x)\right) \to \left(\frac{d\mu}{d{\mathscr{L}^n}}(x),\frac{d\nu}{d{\mathscr{L}^n}}(x)\right)\,\, \text{a.e.}\,x\in {{\mathbf{R}}^n}. \end{align}\] By Fatou’s lemma for the \(L^\infty\) norm, we deduce \[\begin{align} \label{eq95assTop2952} \lVert\frac{d\mu}{d{\mathscr{L}^n}}-\frac{d\nu}{d{\mathscr{L}^n}}\rVert_{L^\infty} \leq \liminf_{j\to\infty} \lVert\frac{d\mu_j}{d{\mathscr{L}^n}}-\frac{d\nu_j}{d{\mathscr{L}^n}}\rVert_{L^\infty}. \end{align}\tag{8}\] Define \[\begin{align} \lambda := \liminf_{j\to \infty} W_\infty(\mu_j,\nu_j). \end{align}\] Then, by 3 , for every \(\varepsilon\ge0\) there exists a subsequence \(({j_k})_{k\geq 1}\) such that \[\begin{align} \mu_{j_k}(A)\leq \nu_{j_k}(A_{\lambda + \varepsilon}) \quad \forall A\in \mathcal{B}({{\mathbf{R}}^n})\quad \forall k\geq 1. \end{align}\] By absolute continuity and \(L^\frac{n}{n-1}\)-convergence of the densities, it follows that \[\begin{align} \label{eq:boh} \mu(B) = \lim_{{k}\to \infty}\mu_{j_k}(B) \leq \lim_{k\to\infty} \nu_{j_k}(B_{\lambda + \varepsilon}) = \nu(B_{\lambda+\varepsilon}) \end{align}\tag{9}\] for every bounded Borel set \(B\in \mathcal{B}({{\mathbf{R}}^n})\). Because \(\text{\normalfont spt}\,\mu \cup \text{\normalfont spt}\,\nu\) is bounded, inequality 9 holds for every Borel set \(B\in \mathcal{B}({{\mathbf{R}}^n})\). This in turn implies \(W_\infty(\mu,\nu) \leq \lambda + \varepsilon\) by 3 . Hence, by arbitrariness of \(\varepsilon\ge0\), we deduce \[\begin{align} \label{eq95assTop2953} W_\infty(\mu,\nu) \leq \liminf_{j\to\infty} W_\infty(\mu_j,\nu_j). \end{align}\tag{10}\] The claim 7 immediately follows from 8 and 10 .

The \(\sigma\)-lower semicontinuity of \(\text{\normalfont Isop}\) is a direct consequence of the lower semicontinuity of the total variation under \(L^1\)-convergence [27]. Therefore, only the second claim requires a proof.

Let \(\lambda \ge0\) and let \((\mu_j)_{j\geq 1}\subseteq\{\text{\normalfont Isop}\leq \lambda\}\) be a \(\mathfrak{d}_\infty^\infty\)-bounded sequence. Let \(K\subseteq {{\mathbf{R}}^n}\) be a regular compact subset such that \(\text{\normalfont spt}\,\mu_j\subseteq K\) for every \(j\geq 1\). Define \(f_j := d\mu_j/d{\mathscr{L}^n}\) and \[\begin{align} u_j := \frac{f_j}{\lVert f_j\rVert_{L^{n/(n-1)}}}\quad \forall j\geq 1. \end{align}\] Then \(\text{\normalfont spt}\,u_j\subseteq K\) and, by Hölder’s inequality \[\begin{align} \lVert u_j\rVert_{\text{\normalfont BV}} = \int_{{\mathbf{R}}^n}u_j\,d{\mathscr{L}^n}+ \lVert\delta u_j\rVert \leq \left({\mathscr{L}^n}(K)\right)^\frac{1}{n} \lVert u_j\rVert_{L^{n/(n-1)}}+ \text{\normalfont Isop}(\mu_j) \leq \left({\mathscr{L}^n}(K)\right)^\frac{1}{n} + \lambda. \end{align}\] Therefore, \((u_j)_{j\geq 1}\subseteq \text{\normalfont BV}(K)\) is a bounded sequence. By virtue of the compact embedding \(\text{\normalfont BV}(K)\hookrightarrow L^1(K)\) (Theorem 4 in Section 5 of [27]), there exists a function \(u_*\in \text{\normalfont BV}(K)\) and a subsequence \((u_{j_k})_{k\geq 1}\preceq (u_j)_{j\geq 1}\) such that \[\begin{align} \lim_{k\to \infty}\lVert u_{j_k}-u_*\rVert_{L^1}=0. \end{align}\] Up to the extraction of a further subsequence and the extension of \(u_{j_k}\) and \(u_*\) by zero in \({{\mathbf{R}}^n}\backslash K\), we can also suppose \[\begin{align} \label{eq95topAss3951} \lim_{k\to\infty} u_{j_k}(x) = u_*(x)\quad \text{a.e.}\,x\in {{\mathbf{R}}^n}. \end{align}\tag{11}\] Observe that \(\mathfrak{d}_\infty^\infty\)-boundedness of \((\mu_j)_{j\geq 1}\) implies boundedness of the sequence \((\lVert f_{j_k}\rVert_{L^{\infty}})_{k\geq 1}\subseteq (0,\infty)\), namely the existence of \(M\ge0\) such that \[\begin{align} \label{eq95topAss3953} \sup_{k\geq 1}\lVert f_{j_k}\rVert_{L^{\infty}}\leq M. \end{align}\tag{12}\] This in turn gives a finite upper bound for the sequence \((\lVert f_{j_k}\rVert_{L^{n/(n-1)}})_{k\geq 1}\subseteq (0,\infty)\). On the other hand, as \(\mu_{j_k}\) is a probability measure that is supported on the compact set \(K\), Hölder’s inequality yields \[\begin{align} \lVert f_{j_k}\rVert_{L^{n/(n-1)}}\geq \left({\mathscr{L}^n}(K)\right)^{-\frac{1}{n}} \end{align}\] for every \(k\geq 1\). Therefore, it is not restrictive to suppose \[\begin{align} \label{eq95topAss3952} \exists L:=\lim_{k\to\infty}\lVert f_{j_k}\rVert_{L^{n/(n-1)}}\in (0,\infty). \end{align}\tag{13}\] Define \(\mu^*:=f^*{\mathscr{L}^n}\), where \(f_*:=Lu_*\). Using 11 and 13 we deduce \(f_{j_k}(x) \to f_*(x)\) as \(k\to\infty\) for almost every \(x\in {{\mathbf{R}}^n}\). Therefore, by 12 and Fatou’s lemma for \(L^\infty\), it follows that \[\begin{align} \label{eq95LpBound2} \lVert f_*\rVert_{L^\infty}\leq M. \end{align}\tag{14}\] Moreover, \[\begin{align} \int_{{\mathbf{R}}^n}f^*\,d{\mathscr{L}^n}= L \int_{{\mathbf{R}}^n}u^*\,d{\mathscr{L}^n}= L\,\lim_{k\to\infty}\frac{\int_{{\mathbf{R}}^n}f_{j_k}\,d{\mathscr{L}^n}}{\lVert f_{j_k}\rVert_{L^{n/(n-1)}}} = 1, \end{align}\] thus \(\mu_*\in \text{\normalfont PL}_\infty^\infty({{\mathbf{R}}^n})\)

Fix an arbitrary \(\varepsilon\ge0\). By Egoroff’s theorem, there exists a measurable subset \(\Omega\subseteq K\) with \({\mathscr{L}^n}(K\backslash\Omega)\le\varepsilon\) such that \((f_{j_k})_{k\geq 1}\) converges uniformly to \(f_*\) in \(\Omega\). Let \(\delta\ge0\) be fixed and let \(k^*:= k^*(\Omega,\delta)\in \mathbf{N}\) be such that \[\begin{align} \sup_{k\geq k^*}\sup_{\Omega} |f_{j_k}-f_*|\leq \delta. \end{align}\] Then, by Hölder’s inequality and the \(L^\infty\)-bounds 12 and 14 , we deduce \[\begin{align} \label{eq95breaks} \begin{aligned} \int_{{{\mathbf{R}}^n}} |f_{j_k}-f_*|^\frac{n}{n-1}\,d{\mathscr{L}^n} &\leq \int_{\Omega}|f_{j_k}-f_*|^\frac{n}{n-1}\,d{\mathscr{L}^n}+ \int_{K\backslash\Omega}\left(|f_{j_k}|+|f^*|\right)^\frac{n}{n-1}\,d{\mathscr{L}^n}\\ &\leq \delta^\frac{n}{n-1}{\mathscr{L}^n}(K) + (2M)^\frac{n}{n-1}\varepsilon \end{aligned} \end{align}\tag{15}\] for every \(k\geq k^*\). Passing to the limit as \(k\to \infty\) and recalling the arbitrariness of \(\varepsilon\), we obtain \[\begin{align} \lim_{k\to\infty} \lVert f_{j_k}-f_*\rVert_{L^{n/(n-1)}} = 0. \end{align}\] Therefore, \(\mu_{j_k}\xrightarrow{\sigma}\mu_*\).

Let \(g\in L^\infty({{\mathbf{R}}^n})\) be a non-negative compactly supported function with finite total variation. Define \(f:=g/\lVert g\rVert_{L^1}\), so that \(\mu_g = f{\mathscr{L}^n}\in \text{\normalfont dom}\,\text{\normalfont Isop}\subseteq {\text{\normalfont PL}}_\infty^\infty({{\mathbf{R}}^n})\). Observe that Proposition 5, Proposition 7 and Proposition 8 guarantee respectively the topological assumptions [topAss1], [topAss2] and [topAss3] for the metric space \(({\text{\normalfont PL}}_\infty^\infty({{\mathbf{R}}^n}),\mathfrak{d}_\infty^\infty)\) endowed with the weak topology \(\sigma\) of the \(L^\frac{n}{n-1}\)-convergence of densities and the functional \(\text{\normalfont Isop}:{\text{\normalfont PL}}_\infty^\infty({{\mathbf{R}}^n})\to[0,\infty]\) defined in 6 . Therefore, Theorem 2 applies and a generalized minimizing movement \(\mu(\cdot)\in\text{\normalfont GMM}(\text{\normalfont Isop};\mu_g)\cap \text{\normalfont AC}^2_\text{\normalfont loc}([0,\infty);\text{\normalfont PL}_\infty^\infty({{\mathbf{R}}^n}))\) exists.

Instead of the isoperimetric functional, one could also consider the \(r\)-Sobolev ratio functional \(\mathcal{S}_r:\text{\normalfont PL}_\infty^{p}({{\mathbf{R}}^n})\to [0,\infty]\) defined by \[\begin{align} \mathcal{S}_r(f{\mathscr{L}^n}):=\frac{\lVert\nabla f\rVert_{L^r}}{\lVert f\rVert_{L^{r^*}}}, \end{align}\] where \(p>r^*\), and prove existence of GMMs for \(\mathcal{S}_r\) in \((\text{\normalfont PL}^p_\infty({{\mathbf{R}}^n}),\mathfrak{d}_\infty^p)\) by adapting the previous proofs.

4 AC curves in the \(\infty\)-Wasserstein space↩︎

We recall and prove the characterization of absolutely continuous curves with respect to \(W_\infty\). Namely, a narrowly continuous curve \(\mu_{(\cdot)}:[0,T]\to \mathcal{P}_\infty({{\mathbf{R}}^n})\) is absolutely continuous with respect to \(W_\infty\) if and only if there exists a Borel vector field \((t,x)\mapsto v_t(x)\) such that \((\mu_{(\cdot)},v_{(\cdot)})\) solves the continuity equation in the sense of Definition 5, and \[\begin{align} \int_0^T\lVert v_t\rVert_{L^\infty(\mu_t)}\,dt<\infty. \end{align}\] Moreover, the metric derivative is exactly the minimal admissible \(L^\infty(\mu_t)\)-norm of such velocity fields.

For \(1<q<\infty\), the corresponding statement is classical and follows from the standard dynamic characterization of absolutely continuous curves in Wasserstein spaces [3], [5], [21]. The endpoint case \(q=\infty\) is well known at a formal level and is often viewed as the limiting case \(q\to\infty\); it is also explicitly mentioned in the literature on dynamical transport distances [5], [22]. However, the passage from finite \(q\) to \(q=\infty\) is not completely trivial. For this reason, and since the endpoint characterization is used in the sequel, we include a self-contained proof. To the best of our knowledge, the precise form needed here is not available in the literature as a standalone statement with proof.

Let us start by proving some basic properties of the \(\infty\)-Wasserstein space.

Recall that a function \(\psi:{{\mathbf{R}}^n}\to \mathbf{R}\cup\{\infty\}\) is lower semicontinuous if and only if the sets \(\{\psi\ge \lambda\}\subseteq {{\mathbf{R}}^n}\) are open for every \(\lambda \in \mathbf{R}\), and that whenever \(\mu_j\) narrowly converges to \(\mu\), then \[\begin{align} \mu(U)\leq \liminf_{j\to\infty}\mu_j(U)\quad \forall U\subseteq {{\mathbf{R}}^n}\text{ open}. \end{align}\]

Let \(\psi:{{\mathbf{R}}^n}\to \mathbf{R}\cup\{\infty\}\) be lower semicontinuous and suppose that \(\mu_j\) narrowly converges to \(\mu\). By contradiction, suppose the existence of a subsequence \((\mu_{j_k})_{k\geq 1}\) such that \[\begin{align} \exists \lim_{k\to\infty}\lVert\psi\rVert_{L^\infty(\mu_{j_k})} \le \lVert\psi\rVert_{L^\infty(\mu)} - \delta, \end{align}\] for some \(\delta \ge 0\). Then there exists \(k^*=k^*(\delta)\in \mathbf{N}\) such that \[\begin{align} \mu_{j_k}\left(\left\{\psi \ge \lVert\psi\rVert_{L^\infty(\mu)}-\frac{\delta}{2}\right\}\right) = 0 \quad \forall k\geq k^*. \end{align}\] Therefore, \[\begin{align} \begin{aligned} 0 \le \mu\left(\left\{\psi \ge \lVert\psi\rVert_{L^\infty(\mu)}-\frac{\delta}{2}\right\}\right) \leq \liminf_{j\to\infty}\mu_{j_k}\left(\left\{\psi \ge \lVert\psi\rVert_{L^\infty(\mu)}-\frac{\delta}{2}\right\}\right) = 0. \end{aligned} \end{align}\]

[Lemma95LSC95i] The families \(\{\mu_j:j\geq 1\},\{\nu_j:j\geq 1\}\subseteq \mathcal{P}({{\mathbf{R}}^n})\) are both tight. This implies that \(\{\gamma_j : j\geq 1\}\subseteq \mathcal{P}({{\mathbf{R}}^n}\times {{\mathbf{R}}^n})\) is tight as well, and by Prokhorov’s theorem we conclude.

[Lemma95LSC95ii] The Euclidean distance \(\text{\normalfont d}_{{\mathbf{R}}^n}(\cdot,\cdot)\) is continuous. In particular, it is lower semicontinuous, therefore Lemma 2 applies and the claim follows.

[Lemma95LSC95iii] This immediately follows from [Lemma95LSC95i] and [Lemma95LSC95ii].

If \((\mathscr{S},\mathfrak{d})\) is a complete metric space and \(\omega:[0,T]\to (\mathscr{S},\text{\normalfont d})\) is an absolutely continuous then there exists a strictly increasing map \(\tau:[0,S]\to [0,T]\) such that \(\omega\circ\tau\) is absolutely continuous and the metric derivative of \(\omega\circ \tau\) is 1 almost everywhere in \([0,S]\). This reparametrization will be called arc-length reparametrization of \(\omega\), and the continuous monotone extension of its inverse \(\sigma:=\tau^{-1}:[0,T]\to [0,S]\) is absolutely continuous and enjoys the property \[\begin{align} \sigma'(t) = |\dot{\omega}|(t)\quad \text{for almost every }0\leq t\leq T \end{align}\] (see e.g. Lemma 1.1.4 of [3])

Clearly \(t\mapsto \mu_t = \nu_{\sigma(t)}\) is narrowly continuous. The integrability condition 4 is an immediate consequence of the change of variables formula for absolutely continuous monotone functions. Therefore, only 5 requires a proof.

Fix a test function \(\varphi\in {{\mathscr{C}}}^\infty_c((0,T)\times {{\mathbf{R}}^n})\) and denote by \(\varphi_t(\cdot)\) the function \(\varphi(t,\cdot)\in {{\mathscr{C}}}^\infty_c({{\mathbf{R}}^n})\). Then, using ?? , the change of coordinates \(s = \sigma(t+r)\) and recalling that there exists \(\delta>0\) such that \(\varphi_t\equiv 0\) for \(t\in [0,T]\backslash[\delta,T-\delta]\), we obtain \[\begin{align} \begin{aligned} \int_0^T\int_{{\mathbf{R}}^n}\partial_t\varphi_t\,d\mu_t\,dt &= \lim_{h\downarrow 0}\int_0^T\int_{{\mathbf{R}}^n}\frac{\varphi_{t}-\varphi_{t-h}}{h}\,d\mu_t\,dt\\ &= -\lim_{h\downarrow 0}\int_0^T\frac{1}{h}\left(\int_{{\mathbf{R}}^n}\varphi_t\,d\nu_{\sigma(t+h)} - \int_{{\mathbf{R}}^n}\varphi_t\,d\nu_{\sigma(t)}\right)\,dt\\ & = - \lim_{h\downarrow 0}\int_0^T\frac{1}{h}\int_{\sigma(t)}^{\sigma(t+h)}\int_{{\mathbf{R}}^n}\langle\nabla \varphi_t, w_s\rangle\,d\nu_{s}\,ds\,dt\\ &= - \lim_{h\downarrow 0}\int_0^T\frac{1}{h} \int_0^{h}\int_{{\mathbf{R}}^n}\langle \nabla \varphi_t,v_{t+r}\rangle\,d\mu_{t+r}\,dr\,dt\\ &= - \lim_{h\downarrow 0} \frac{1}{h}\int_0^h \int_0^T \int_{{\mathbf{R}}^n}\langle \nabla \varphi_{t-r},v_{t}\rangle \,d\mu_{t}\,dt\,dr\\ &= - \int_0^T\int_{{\mathbf{R}}^n}\langle \nabla \varphi_t,v_t\rangle \,d\mu_t\,dt, \end{aligned} \end{align}\] which is exactly 5 . The claim then follows by arbitrariness of the choice of \(\varphi\).

We say that \(\{\rho_\varepsilon: \varepsilon\ge 0\}\) is a family of strictly positive rapidly decreasing mollifiers if \(\rho\in {{\mathscr{C}}}^\infty_0({{\mathbf{R}}^n})\cap L^1({{\mathbf{R}}^n})\) is a positive radially symmetric function with \(\int_{{\mathbf{R}}^n}\rho {\mathscr{L}^n}=1\), \(x\mapsto |x|^N\rho(x)\) belongs to \(L^1({{\mathbf{R}}^n})\) for every \(N\geq 0\) and \(\rho_\varepsilon= \varepsilon^{-n}\rho(\cdot/\varepsilon)\) for every \(\varepsilon\ge 0\).

If \(M\in \mathcal{M}({{\mathbf{R}}^n};\mathbf{R}^{m})\) is a \(m\)-valued Borel measure and \(\rho\in {{\mathscr{C}}}^\infty_0({{\mathbf{R}}^n})\cap L^1({{\mathbf{R}}^n})\), we define the convolution of \(M\) by \(\rho\) as the function \(M\Asterisk\rho:{{\mathbf{R}}^n}\to \mathbf{R}^{m}\) \[\begin{align} M\Asterisk\rho (x):=\int_{{{\mathbf{R}}^n}} \rho(x-y)\,dM(y)\quad \forall x\in {{\mathbf{R}}^n}. \end{align}\]

It is easy to verify that, under the standing assumptions, \((\mu_{(\cdot)}^\varepsilon,v_{(\cdot)}^\varepsilon)\) is a weak solution of the continuity equation in the sense of Definition 5.

The proofs of properties ?? , ?? and ?? are done exactly as in Lemma 8.1.9 of [3]. Therefore, it is enough to prove ?? , and it is enough to observe that for almost every \(0 \leq t \leq T\) we have \[\begin{align} |v_t^\varepsilon(x)| = \frac{|E_t^\varepsilon(x)|}{f_t^\varepsilon(x)} \leq \frac{\int_{{\mathbf{R}}^n}\rho_\varepsilon(x-y)|v_t(y)|\,d\mu_t(y)}{\int_{{\mathbf{R}}^n}\rho_\varepsilon(x-y)\,d\mu_{t}(y)}\leq \lVert v_t\rVert_{L^\infty(\mu_t)}\quad \forall x\in {{\mathbf{R}}^n}. \end{align}\]

Thanks to Lemma 5, there are approximations \(\{(\mu_{(\cdot)}^\varepsilon,v_{(\cdot)}^\varepsilon) : \varepsilon\ge 0\}\) that are regular solutions of the continuity equation. Therefore, using Proposition 8.1.8 of [3], if \((t,x)\mapsto T_t^\varepsilon(x)\) is the flow-map associated with the time-dependent vector field \((t,x)\mapsto v_t^\varepsilon(x)\), then \(T^\varepsilon\) is defined in \([0,T]\times {{\mathbf{R}}^n}\) and \(\mu_t^\varepsilon= (T_t^\varepsilon)_\#\mu_0^\varepsilon\) for every \(0\leq t\leq T\).

Fix \(x\in {{\mathbf{R}}^n}\) and \(0\leq t_1 \le t_2 \leq T\). Then, by definition of flow-map and ?? , \[\begin{align} |T_{t_2}^\varepsilon(x)-T_{t_1}^\varepsilon(x)| \leq \int_{t_1}^{t_2}|\dot{T}_s^\varepsilon(x)|\,ds \leq \int_{t_1}^{t_2}\lVert v_s\rVert_{L^\infty(\mu_s)}\,ds. \end{align}\] Since \(\gamma_{t_1,t_2}^\varepsilon=(T_{t_2}^\varepsilon\times T_{t_1}^\varepsilon)_\#\mu_0^\varepsilon\) is a coupling of \(\mu_{t_1}^\varepsilon\) and \(\mu_{t_2}^\varepsilon\), it follows that \[\begin{align} \label{eq95oneIneq951} W_\infty(\mu_{t_1}^\varepsilon,\mu_{t_2}^\varepsilon)\leq \lVert\text{\normalfont d}_{{\mathbf{R}}^n}(\cdot,\cdot)\rVert_{L^\infty(\gamma_{t_1,t_2}^\varepsilon)}\leq \int_{t_1}^{t_2}\lVert v_s\rVert_{L^\infty(\mu_s)}\,ds. \end{align}\tag{16}\] Take the inferior limit of 16 as \(\varepsilon\downarrow 0\) and combine the narrow convergence result ?? and the lower semicontinuity of \(W_\infty\) under narrow convergence given by Lemma 3 to obtain \[\begin{align} W_\infty(\mu_{t_1},\mu_{t_2})\leq \int_{t_1}^{t_2}\lVert v_s\rVert_{L^\infty(\mu_s)}\,ds. \end{align}\] This ends the proof.

By virtue of Lemma 4, it is not restrictive to suppose \(|\dot{\mu}|^{W_\infty}\equiv 1\) almost everywhere in \([0,T]\).

Fix a test function \(\psi\in {{\mathscr{C}}}^\infty_c({{\mathbf{R}}^n})\) and consider the function \(t\mapsto \mu^\psi(t):=\int_{{\mathbf{R}}^n}\psi\,d\mu_t\). For any \(0\leq s \le t\leq T\), let \(\gamma_{s,t}\in \Gamma_\infty(\mu_s,\mu_t)\) be an optimal coupling, and denote by \(H_\psi:{{\mathbf{R}}^n}\times{{\mathbf{R}}^n}\to [0,\infty)\) the function defined by \[\begin{align} H_\psi(x,y):=\begin{cases} |\nabla \psi(x)| &,\text{ if }x=y,\\ \text{ }\\ \frac{|\psi(x)-\psi(y)|}{|x-y|} &,\text{ if }x\neq y \end{cases}. \end{align}\] Then, for any fixed \(0\le t\le T\) and \(h \ge 0\) such that \(0\le t+h \le T\), we have \[\begin{align} \begin{aligned} \frac{|\mu^\psi(t+h)-\mu^\psi(t)|}{|h|} &\leq \frac{1}{|h|}\int_{{{\mathbf{R}}^n}\times {{\mathbf{R}}^n}} |x-y| H_\psi(x,y)\,d\gamma_{t+h,t}(x,y)\\ &\leq \frac{W_\infty(\mu_{t+h},\mu_t)}{|h|}\int_{{{\mathbf{R}}^n}\times{{\mathbf{R}}^n}} H_\psi(x,y)\,d\gamma_{t+h,t}(x,y). \end{aligned} \end{align}\] Observe that, by Lemma 3, for every \(0\le t \le T\), \(\gamma_{t+h,t}\) converges narrowly to \((\text{\normalfont id}\times \text{\normalfont id})_\#\mu_t\) as \(h\to 0\). Moreover, the function \(H_\psi\) is upper semicontinuous. Therefore, for any \(0\le t\le T\) such that the metric derivative \(|\dot{\mu}|^{W_\infty}(t)\) exists and equals 1, we have \[\begin{align} \label{eq95ineqB951} \limsup_{h\to 0} \frac{|\mu^\psi(t+h)-\mu^\psi(t)|}{|h|} \leq \lVert\nabla \psi\rVert_{L^1(\mu_t)}. \end{align}\tag{17}\]

Define the positive measure in \(\widetilde{\mu}:= {\mathscr{L}}^1{\mathrel{\makebox[7pt][c]{\rule{0.4pt}{6.75pt}\rule{5.5pt}{.4pt}}}}[0,T]\otimes (\mu_t)_t\in \mathcal{M}_+([0,T]\times {{\mathbf{R}}^n})\). Then, for every smooth test function \(\varphi\in {{\mathscr{C}}}^\infty_c((0,T)\times {{\mathbf{R}}^n})\) we have \[\begin{align} \begin{aligned} \int_{[0,T]\times {{\mathbf{R}}^n}} \partial_s \varphi(s,x)\,d\widetilde{\mu}(s,x) &= \lim_{h\downarrow 0}\int_{[0,T]\times {{\mathbf{R}}^n}} \frac{\varphi(s,x)-\varphi(s-h,x)}{h}\,d\widetilde{\mu}(s,x)\\ &= \lim_{h\downarrow 0} \int_{[0,T]} \frac{\mu^{\varphi(s,\cdot)}(s)- \mu^{\varphi(s,\cdot)}(s+h)}{h}\,ds. \end{aligned} \end{align}\] Hence, recalling 17 , \[\begin{align} \label{eq95ineqB952} \begin{aligned} |\int_{[0,T]\times {{\mathbf{R}}^n}} \partial_s \varphi(s,x)\,d\widetilde{\mu}(s,x)| &\leq \int_{[0,T]} \lVert\nabla \varphi(s,\cdot)\rVert_{L^1(\mu_s)}\,ds. \end{aligned} \end{align}\tag{18}\] Consider the linear subspace \[\begin{align} V:=\left\{\nabla \varphi: \varphi\in {{\mathscr{C}}}^\infty_c((0,T)\times {{\mathbf{R}}^n})\right\}\subseteq L^1(\widetilde{\mu};{{\mathbf{R}}^n}) \end{align}\] endowed with the restriction of the \(L^1(\widetilde{\mu})\)-norm and the linear functional \(L:V\to \mathbf{R}\) \[\begin{align} L(\nabla \varphi):=-\int_{[0,T]\times{{\mathbf{R}}^n}}\partial_s\varphi(s,x)\,d\widetilde{\mu}(s,x). \end{align}\] By 18 , \(L\) is a bounded linear operator with \(\text{\normalfont Lip}(L)\leq 1\). Therefore, by Hahn-Banach, \(L\) can be extended to a functional \(\tilde{L}:L^1(\widetilde{\mu};{{\mathbf{R}}^n})\to \mathbf{R}\) with \(\text{\normalfont Lip}(\tilde{L}) = \text{\normalfont Lip}(L)\leq 1\). By Riesz representation theorem of the dual of \(L^1\), there exists one unique element \(v\in L^\infty(\widetilde{\mu};{{\mathbf{R}}^n})\) such that \[\begin{gather} \tag{19} \tilde{L}(w)= \int_{[0,T]\times {{\mathbf{R}}^n}}\langle v , w\rangle \,d\widetilde{\mu}\quad \forall w\in L^1(\widetilde{\mu};{{\mathbf{R}}^n}),\\ \lVert v\rVert_{L^\infty(\widetilde{\mu})} = \text{\normalfont Lip}(\tilde{L})\leq 1.\tag{20} \end{gather}\]

Define \(v_t := v(t,\cdot)\) for \({\mathscr{L}}^1\)-almost every \(0\leq t\leq T\). Then 19 trivially gives ?? and ?? , and 20 yields ?? . This ends the proof.

Combining Theorem 4, Theorem 10 and Theorem 11, one obtains the following characterization of absolute continuity in \((\mathcal{P}_q({{\mathbf{R}}^n}),W_q)\) for every \(1<q\leq \infty\) in terms of solutions of the continuity equation, that we formulate as the following unified corollary.

5 Absolutely continuous curves in \({\text{\normalfont PL}}_q^p(\mathbf{R}^n)\)↩︎

Throughout this section any curve of probability measures \(\mu_{(\cdot)}\) will be implicitly supposed to be defined on a fixed interval \([0,T]\), for some \(T>0\), to take values in either \(\mathcal{P}_q({{\mathbf{R}}^n})\) or \({\text{\normalfont PL}_q^p({{\mathbf{R}}^n})}\), and to be narrowly continuous.

As in this section more than one notion of absolute continuity for a curve \(\mu_{(\cdot)}\) will be considered, we introduce the following convention. We shall say that \(\mu_{(\cdot)}\) is \(W_q\)-absolutely continuous if \(\mu_{(\cdot)}\in \text{\normalfont AC}([0,T];\mathcal{P}_q({{\mathbf{R}}^n}))\). Analogously, we say that \(\mu_{(\cdot)}\) is \({\mathfrak{d}_q^p}\)-absolutely continuous \(\mu_{(\cdot)}\in \text{\normalfont AC}([0,T];{\text{\normalfont PL}_q^p({{\mathbf{R}}^n})})\). The notation \(|\dot{\mu}|(\cdot)\) will be exclusively used for denoting the \({\mathfrak{d}_q^p}\)-metric derivative of \(\mu_{(\cdot)}\), while its \(W_q\)-metric derivative will be denoted by \(|\dot{\mu}|^{W_q}(\cdot)\). In a similar way, the \(L^p\).metric derivative of a curve \(f_{(\cdot)}:[0,T]\to L^p({{\mathbf{R}}^n})\), whenever it is defined, is denoted by \(|\dot{f}|^{L^p}\).

5.1 Absolutely continuous curves in \(\text{\normalfont PL}_q^p({{\mathbf{R}}^n})\) for \(1<p\leq \infty\).↩︎

Let us introduce the following notation. We say that a Borel function \(F:[0,T]\times {{\mathbf{R}}^n}\to\mathbf{R}\) is \(L^1\) in time and \(L^p\) in space, and write \(F\in L^1_tL^p_x([0,T]\times {{\mathbf{R}}^n})\) if for almost every \(0\leq t\leq T\) the \(t\)-section \(F_t:=F(t,\cdot)\) belongs to \(L^p({{\mathbf{R}}^n})\) and \[\begin{align} \int_0^T\lVert F_t\rVert_{L^p}\,dt<\infty. \end{align}\]

If \(\mu_{(\cdot)}=f_{(\cdot)}{\mathscr{L}^n}:[0,T]\to {\text{\normalfont PL}_q^p({{\mathbf{R}}^n})}\) is \({\mathfrak{d}_q^p}\)-absolutely continuous, then \[\begin{align} {W_q(\mu_{t},\mu_s)}+ {\lVert f_{t}-f_s\rVert}_{L^p} = {\mathfrak{d}_q^p}(\mu_{t},\mu_s) \leq \int_s^t |\dot{\mu}|(r)\,dr\quad \forall 0\leq s \le t \leq T. \end{align}\] Therefore \(\mu_{(\cdot)}\) is \(W_q\)-absolutely continuous and \(f_{(\cdot)}\) is \(L^p\)-absolutely continuous. Viceversa, if \(\mu_{(\cdot)}\) is \(W_q\)-absolutely continuous and \(f_{(\cdot)}\) is \(L^p\)-absolutely continuous, then \[\begin{align} {\mathfrak{d}_q^p}(\mu_{t},\mu_s) = {W_q(\mu_{t},\mu_s)}+ {\lVert f_{t}-f_s\rVert}_{L^p} \leq \int_s^t (|\dot{\mu}|^{W_q}(r) + |\dot{f}|^{L^p}(r))\,dr. \end{align}\] This proves the first claim. Moreover, by the very definition of the metric derivative, it follows that if \(\mu_{(\cdot)}\) is \({\mathfrak{d}_q^p}\)-absolutely continuous, then \[\begin{align} \label{eq95th95metricDerivatives} |\dot{\mu}|(t) = |\dot{\mu}|^{W_q}(t) + |\dot{f}|^{L^p}(t)\quad \text{for almost every }0\leq t\leq T. \end{align}\tag{21}\]

Let us now fix a \({\mathfrak{d}_q^p}\)-absolutely continuous curve \(\mu_{(\cdot)}\). Recalling Lemma 4, it is enough to prove the statement for a curve \(\mu_{(\cdot)}\) parametrized by arc-length, i.e. with \(|\dot{\mu}|(t)=1\) at almost every \(0\leq t\leq T\). By virtue of Theorem 4, if \(1<q<\infty\), or Corollary 2, if \(q=\infty\), there exists a Borel vector field \((t,x)\mapsto v_{t}(x)\) such that \[\begin{gather} \int_0^T\int_{{\mathbf{R}}^n}(\partial_t\varphi_t + \langle \nabla \varphi_t,v_t \rangle) f_t\,d{\mathscr{L}^n} \,dt = 0 \quad \forall \varphi\in {{\mathscr{C}}}^\infty_c((0,T)\times {{\mathbf{R}}^n}),\tag{22}\\ \lVert v_t\rVert_{L^q(\mu_t)} = |\dot{\mu}|^{W_q}(t) \leq 1\quad \text{for almost every }0\leq t \leq T.\tag{23} \end{gather}\]

Taking into account Lemma 1, as \(f_{(\cdot)}\) is \(L^p\)-absolutely continuous with essentially bounded \(L^p\)-metric derivative, there exists a function \(\mathcal{D}f_{(\cdot)}\in L^\infty([0,T];L^p({{\mathbf{R}}^n}))\), if \(1<p<\infty\), or \(\mathcal{D}f_{(\cdot)}\in L_{w^*}^\infty([0,T];L^\infty({{\mathbf{R}}^n}))\), if \(p=\infty\), such that \[\begin{gather} \lim_{h\to 0}\int_{{{\mathbf{R}}^n}} \psi \frac{f_{t+h}-f_{t}}{h}\,d{\mathscr{L}^n}= \int_{{{\mathbf{R}}^n}} \psi \mathcal{D}f_t\,d{\mathscr{L}^n}\,\,\forall \psi \in {{\mathscr{C}}}^0_0({{\mathbf{R}}^n})\,\text{a.e. }0\leq t\leq T,\tag{24}\\ f_b-f_a = \int_{a}^b \mathcal{D}f_t\,dt \quad\text{in }L^p({{\mathbf{R}}^n}),\,\forall 0\le a\leq b \le T,\tag{25}\\ \lVert\mathcal{D}f_t\rVert_{L^p}= |\dot{f}|(t)\leq 1 \quad \text{for a.e. }0\leq t\leq T.\tag{26} \end{gather}\]

Fix a smooth compactly supported function \(\varphi\in {{\mathscr{C}}}^\infty_c((0,T)\times{{\mathbf{R}}^n})\) and, for every \(0\leq t \leq T\), denote by \(\varphi_t\in {{\mathscr{C}}}^\infty_c({{\mathbf{R}}^n})\) the function \(x\mapsto \varphi(t,x)\). Then \[\begin{align} \begin{aligned} \int_0^T\int_{{\mathbf{R}}^n}\partial_t \varphi_t f_t\,d{\mathscr{L}^n}\,dt &= \lim_{h\downarrow 0}\int_{{\mathbf{R}}^n}\int_0^T \frac{\varphi_t - \varphi_{t-h}}{h}f_t\,dt\,d{\mathscr{L}^n}\\ &= \lim_{h\downarrow 0}\int_{{{\mathbf{R}}^n}}\frac{1}{h}\left(\int_0^T \varphi_t f_t\,dt - \int_0^T \varphi_{t-h}f_t\,dt\right)\,d{\mathscr{L}^n}. \end{aligned} \end{align}\] Since there exists \(\delta \ge 0\) and a compact \(K\subseteq {{\mathbf{R}}^n}\) such that \(\text{\normalfont spt}\varphi\subseteq [\delta, T-\delta]\times K\), we can apply the change of variables \(t\mapsto t-h\) to the second integral to obtain \[\begin{align} \label{eq95ACPinftp951} \int_0^T \int_{{\mathbf{R}}^n}\partial_t\varphi_t f_t\,d{\mathscr{L}^n}\,dt = -\lim_{h\downarrow 0}\int_0^T\int_{{\mathbf{R}}^n}\varphi_t\frac{f_{t+h}-f_t}{h}\,d{\mathscr{L}^n}\,dt. \end{align}\tag{27}\] From 24 , we deduce that \[\begin{align} \lim_{h\downarrow 0}\int_{{\mathbf{R}}^n}\varphi_t \frac{\,f_{t+h}-f_t}{h}\,d{\mathscr{L}^n}= \int_{{\mathbf{R}}^n}\varphi_t \mathcal{D}f_t\,d{\mathscr{L}^n}\quad \text{for a.e. }0\leq t \leq T. \end{align}\] Since \[\begin{align} \left|\int_{{\mathbf{R}}^n}\varphi_t \frac{\,f_{t+h}-f_t}{h}\,d{\mathscr{L}^n}\right| \leq \frac{\lVert\varphi\rVert_{{{\mathscr{C}}}^0}}{h}\int_t^{t+h}\int_K |\mathcal{D}f_s|\,d{\mathscr{L}^n}\,ds \leq \lVert\varphi\rVert_{{{\mathscr{C}}}^0}\left({\mathscr{L}^n}(K)\right)^\frac{1}{p'}, \end{align}\] where \(1\leq p'<\infty\) is the Hölder conjugate of \(p\), by dominated convergence we deduce that \[\begin{align} \label{eq95ACPinftp952} \lim_{h\downarrow 0} \int_0^T\int_{{\mathbf{R}}^n}\varphi_t \frac{\,f_{t+h}-f_t}{h}\,d{\mathscr{L}^n}= \int_0^T \int_{{\mathbf{R}}^n}\varphi_t \mathcal{D}f_t\,d{\mathscr{L}^n}\,dt. \end{align}\tag{28}\] Combining 22 , 27 and 28 , we obtain \[\begin{align} \label{eq95ACPinftyp953} -\int_0^T\int_{{\mathbf{R}}^n}\langle \nabla \varphi_t,v_t f_t\rangle\,d{\mathscr{L}^n}\,dt = \int_0^T \int_{{\mathbf{R}}^n}\partial_t\varphi_t f_t\,d{\mathscr{L}^n}\,dt = - \int_0^T \int_{{\mathbf{R}}^n}\varphi_t \partial f_t\,d{\mathscr{L}^n}\,dt. \end{align}\tag{29}\] Equation 29 , together with 26 , proves [th95ACPinftyp95i], [th95ACPinftyp95ii] and [th95ACPinftyp95iibis]. Finally, 21 , 23 , and 26 imply [th95ACPinftyp95iv].

Suppose now \(\mu_{(\cdot)}=f_{(\cdot)}{\mathscr{L}^n}:[0,T]\to \text{\normalfont PL}_q^p({{\mathbf{R}}^n})\) to be narrowly continuous and the existence of a Borel vector field \((t,x)\mapsto v_t(x)\) such that properties [th95ACPinftyp950]–[th95ACPinftyp95iiii] hold. Then \(v\) satisfies the integrability condition 4 of Definition 5 and, combining [th95ACPinftyp95i], [th95ACPinftyp95ii] and [th95ACPinftyp95iibis], it immediately follows that \((\mu_{(\cdot)},v_{(\cdot)})\) is a weak solution of the continuity equation in \([0,T]\). Thus, from Theorem 4 (resp. Theorem 10, if \(q=\infty\)), we deduce that \(\mu_{(\cdot)}\) is \(W_q\)-absolutely continuous and that \[\begin{align} \label{E951} |\dot{\mu}|^{W_q}(t)\leq \lVert v_t\rVert_{L^q(\mu_t)} \quad \text{for almost every }0\leq t\leq T. \end{align}\tag{30}\]

Fix \(0\leq t\leq t+h\leq T\). We show that \[\begin{align} \label{eqq951} f_{t+h} = f_t - \int_t^{t+h}\text{\normalfont div}(v_\tau f_\tau)\,d\tau\quad \text{in }L^p({{\mathbf{R}}^n}), \end{align}\tag{31}\] where the integral in the right-hand side is the Bochner integral (resp. the weak\(^*\) integral, if \(p=\infty\)) of \(\tau \mapsto \text{\normalfont div}(v_\tau f_\tau)\). First, observe that – thanks to [th95ACPinftyp95i] and [th95ACPinftyp95iiii] – \(\tau \mapsto \text{\normalfont div}(v_\tau f_\tau)\) is a Bochner measurable (resp. weakly\(^*\) measurable, if \(p=\infty\)) curve \([0,T]\to L^p({{\mathbf{R}}^n})\), and it belongs to \(L^1([0,T];L^p({{\mathbf{R}}^n}))\) (resp. \(L^1_{w^*}([0,T];L^\infty({{\mathbf{R}}^n}))\), if \(p=\infty\)). Therefore, the right-hand side of 31 is well-defined.

We shall now prove the claim 31 . It will be enough to show that \[\begin{align} \label{eqq953} \int_{{\mathbf{R}}^n}\left(f_{t+h} - f_t + \int_t^{t+h}\text{\normalfont div}(v_\tau f_\tau)\,d\tau\right)g\,d{\mathscr{L}^n}= 0\quad \forall g\in {{\mathscr{C}}}^\infty_c({{\mathbf{R}}^n}). \end{align}\tag{32}\] Fix a smooth test function \(g\in {{\mathscr{C}}}^\infty_c({{\mathbf{R}}^n})\). Then, recalling Remark 3, the function \(\tau\mapsto \int_{{\mathbf{R}}^n}f_\tau g\,d{\mathscr{L}^n}\) is absolutely continuous, and in particular \[\begin{align} \int_{{\mathbf{R}}^n}f_{t+h}g\,d{\mathscr{L}^n}- \int_{{\mathbf{R}}^n}f_tg\,d{\mathscr{L}^n}= \int_t^{t+h}\int_{{\mathbf{R}}^n}\langle \nabla g,v_\tau\rangle\,d{\mathscr{L}^n}\,d\tau. \end{align}\] Therefore, \[\begin{align} \label{eqq952} \begin{aligned} \int_{{\mathbf{R}}^n}&\left(f_{t+h} - f_t + \int_t^{t+h}\text{\normalfont div}(v_\tau f_\tau)\,d\tau\right)g\,d{\mathscr{L}^n} = \\ &\quad\quad\int_0^{T}\int_{{\mathbf{R}}^n}\left(\langle \nabla g\,\chi_{[t,t+h]}(\tau),v_\tau\rangle + \text{\normalfont div}(v_\tau f_\tau)g\,\chi_{[t,t+h]}(\tau)\right)\,d\tau\,d{\mathscr{L}^n}. \end{aligned} \end{align}\tag{33}\] Approximating \(\chi_{[t,t+h]}\) with a smooth function that is compactly supported in \([0,T]\) and integrating by parts, we prove that the right-hand side of 33 is zero. By arbitrariness of the choice of \(g\), 32 follows.

Using 31 and the triangle inequality for integrals, we can write \[\begin{align} \lVert f_{t+h}-f_t\rVert_{L^p}\leq \int_t^{t+h}\lVert\text{\normalfont div}(v_\tau f_\tau)\rVert_{L^p}\,d\tau. \end{align}\] Therefore, by arbitrariness of \(t\) and \(h\), and taking into account [th95ACPinftyp95iiii], \(f_{(\cdot)}\) is \(L^p\)-absolutely continuous and \[\begin{align} \label{E952} |\dot{f}|^{L^p}(t)\leq \lVert\text{\normalfont div}(v_t f_t)\rVert_{L^p}\quad \text{for almost every }0\leq t\leq T. \end{align}\tag{34}\]

Combining 30 and 34 , we deduce that \(\mu_{(\cdot)}\) is \({\mathfrak{d}_q^p}\)-absolutely continuous and that ?? holds true.

5.2 Absolutely continuous curves in \(\text{\normalfont PL}_q^1({{\mathbf{R}}^n})\).↩︎

We say that a Borel measure \(M = {\mathscr{L}}^1{\mathrel{\makebox[7pt][c]{\rule{0.4pt}{6.75pt}\rule{5.5pt}{.4pt}}}}[0,T]\otimes (M_t)_{t}\in \mathcal{M}([0,T]\times {{\mathbf{R}}^n})\) is \(\mathcal{M}^1\) in time and of finite variation in space, and write \(M\in \mathcal{M}^1_t\text{\normalfont FV}_x([0,T]\times {{\mathbf{R}}^n})\) if \[\begin{align} \int_0^T\lVert M_t\rVert_{\mathrm{\small tv}}\,dt<\infty. \end{align}\]

The first claim is proved exactly as in Theorem 13.

Fix a curve \(\mu_{(\cdot)}=f_{(\cdot)}{\mathscr{L}^n}\in \text{\normalfont AC}([0,T];\text{\normalfont PL}_q^1({{\mathbf{R}}^n}))\). Consider the isometric embedding of \((L^1({{\mathbf{R}}^n}),\lVert\cdot\rVert_{L^1})\) into the space of signed real measures \((\mathcal{M}({{\mathbf{R}}^n}),\lVert\cdot\rVert_{\mathrm{\small tv}})\simeq ({{\mathscr{C}}}^0_b({{\mathbf{R}}^n}),\lVert\cdot\rVert_{{{\mathscr{C}}}^0})^*\) given by \(f\hookrightarrow f{\mathscr{L}^n}\). Therefore, the curve \(f_{(\cdot)}\in \text{\normalfont AC}^\infty([0,T];L^1({{\mathbf{R}}^n}))\) is identified (via the aforementioned embedding) with the curve \(f_{(\cdot)}{\mathscr{L}^n}\in \text{\normalfont AC}^\infty([0,T];\mathcal{M}({{\mathbf{R}}^n}))\). Hence, by virtue of Lemma 1, there exists a curve \(\mathcal{D}f_{(\cdot)}\in L^\infty_{w^*}([0,T];\mathcal{M}({{\mathbf{R}}^n}))\) such that \[\begin{gather} \lim_{h\to 0}\int_{{{\mathbf{R}}^n}} \psi \frac{f_{t+h}-f_{t}}{h}\,d{\mathscr{L}^n}= \int_{{{\mathbf{R}}^n}} \psi\, d\mathcal{D}f_t \quad\forall \psi \in {{\mathscr{C}}}^0_0({{\mathbf{R}}^n})\,\text{ for a.e. }0\leq t\leq T,\\ \int_{{\mathbf{R}}^n}\psi\,(f_b-f_a)\,d{\mathscr{L}^n}= \int_{a}^b\int_{{\mathbf{R}}^n}\psi\, d\mathcal{D}f_t\,dt \quad\forall \psi \in {{\mathscr{C}}}^0_0({{\mathbf{R}}^n})\,\forall 0\le a\leq b \le T,\\ \lVert\mathcal{D}f_t\rVert_{\mathrm{\small tv}}= |\dot{f}|(t)\leq 1 \quad \text{for a.e. }0\leq t\leq T. \end{gather}\]

Fix a test function \(\varphi\in {{\mathscr{C}}}^\infty_c([0,T]\times {{\mathbf{R}}^n})\). Arguing as in the proof of Theorem 13, we obtain the equality \[\begin{align} \int_0^T\int_{{\mathbf{R}}^n}\partial_t\varphi_t f_t\,d{\mathscr{L}^n}\,dt = -\lim_{h\downarrow 0}\int_0^T\int_{{\mathbf{R}}^n}\varphi_t\frac{f_{t+h}-f_t}{h}\,d{\mathscr{L}^n}\,dt, \end{align}\] and the limit \[\begin{align} \lim_{h\downarrow 0} \int_{{\mathbf{R}}^n}\varphi_t\frac{f_{t+h}-f_t}{h}\,d{\mathscr{L}^n}= \int_{{\mathbf{R}}^n}\varphi_t\,d\mathcal{D}f_t\quad \text{for almost every }0\leq t\leq T. \end{align}\] To justify the exchange of limit and integral, it is enough to observe that \[\begin{align} \left|\int_{{\mathbf{R}}^n}\varphi_t \frac{\,f_{t+h}-f_t}{h}\,d{\mathscr{L}^n}\right| \leq \frac{\lVert\varphi\rVert_{{{\mathscr{C}}}^0}}{h}\int_t^{t+h}\lVert\mathcal{D}f_s\rVert_{\mathrm{\small tv}}\,ds \leq \lVert\varphi\rVert_{{{\mathscr{C}}}^0} \end{align}\] and to use the dominated convergence theorem. Thus, if \(\mu_{(\cdot)}\) is \({\mathfrak{d}_q^1}\)-absolutely continuous and \(v_{(\cdot)}\) is the velocity field of \(\mu_{(\cdot)}\) given by Theorem 4, for \(1<q<\infty\), or by Corollary 2, if \(q=\infty\), then \[\begin{align} -\int_0^T\int_{{\mathbf{R}}^n}\langle \nabla \varphi_t,v_t\rangle f_t\,d{\mathscr{L}^n}\,dt = \int_0^T\int_{{\mathbf{R}}^n}\partial_t\varphi_t\,f_t\,d{\mathscr{L}^n}= -\int_0^T\int_{{\mathbf{R}}^n}\varphi_t\,d\mathcal{D}f_t\,dt, \end{align}\] and properties [th95ACPinfty1950]–[th95ACPinfty195iv] are proved.

Suppose now \(\mu_{(\cdot)}=f_{(\cdot)}{\mathscr{L}^n}:[0,T]\to {\text{\normalfont PL}_q^1({{\mathbf{R}}^n})}\) is narrowly continuous and there exists a Borel time-dependent vector field \((t,x)\mapsto v_t(x)\) such that properties [th95ACPinfty1950]–[th95ACPinfty195iiii] hold. Then, arguing as in Theorem 13 one proves that \((\mu_{(\cdot)},v_{(\cdot)})\) is a weak solution of the continuity equation in \([0,T]\). Therefore \(\mu_{(\cdot)}\) is \(W_q\)-absolutely continuous and the estimate \[\begin{align} |\dot{\mu}|^{W_q}(t)\leq \lVert v\rVert_{L^q(\mu_t)} \end{align}\] holds for almost every \(0\leq t\leq T\). On the other hand, using the absolute continuity of the map \(\tau\mapsto \int_{{\mathbf{R}}^n}g\,d\mu_\tau\) given by Remark 3, for every \(g\in {{\mathscr{C}}}^\infty_c({{\mathbf{R}}^n})\), together with a smooth compactly supported approximation of \(\chi_{[t,t+h]}\), one proves that \[\begin{align} \mu_{t+h} = \mu_t + \int_t^{t+h}\text{\normalfont div}(v_\tau f_\tau)\,d\tau\quad \text{in }\mathcal{M}({{\mathbf{R}}^n}) \end{align}\] for all \(0\leq t\leq t+h\leq T\). Therefore, \[\begin{align} \lVert f_{s}-f_t\rVert_{L^1} = \lVert\mu_{s}-\mu_t\rVert_{\mathrm{\small tv}}\leq \int_t^{s}\lVert\text{\normalfont div}(v_\tau f_\tau)\rVert_{\mathrm{\small tv}}\,d\tau\quad \forall 0\leq t\leq s\leq T. \end{align}\] Hence \(f_{(\cdot)}\) is \(L^1\)-absolutely continuous and ?? holds.

5.3 Examples of \({\mathfrak{d}_q^p}\)-AC curves.↩︎

Theorem 13 and Theorem 14 give a complete characterization of \({\mathfrak{d}_q^p}\)-absolutely continuous curves in terms of solution of the continuity equations for velocity fields \(v_{(\cdot)}\) that satisfy a Sobolev-like condition involving the \(L^q(\mu_t)\)-norm of \(v_t\) and the \(L^p({{\mathbf{R}}^n})\)-norm of \(\text{\normalfont div}(vf)\). However, Theorems 13 and 14 do not guarantee actual existence of such fields, nor of \(\mathfrak{d}_q^p\)-absolutely continuous curves.

We shall now give some examples of \({\mathfrak{d}_q^p}\)-absolutely continuous curves.

6 Final remarks and open questions↩︎

We conclude the paper by collecting some natural questions and possible directions for further research. We hope that these problems may be of interest to other researchers and may contribute to the further development of the metric framework introduced in this paper.

This paper shows that the minimizing movement scheme for the isoperimetric functional admits generalized minimizing movements which are absolutely continuous in \(\text{\normalfont PL}_q^p({{\mathbf{R}}^n})\). We also obtained a characterization of absolutely continuous curves in \(\text{\normalfont PL}_q^p({{\mathbf{R}}^n})\) in terms of the continuity equation and suitable Eulerian estimates.

These results do not imply, however, that \(\text{\normalfont PL}_q^p({{\mathbf{R}}^n})\) contains many non-constant absolutely continuous curves. The characterization theorem is a criterion, not an existence result for curves with prescribed endpoints. In principle, it could happen that the metric \(\mathfrak{d}_q^p\) is so restrictive that the only absolutely continuous curves starting from certain measures are constant curves. In such a situation, the GMMs constructed by the scheme with these initial data would be dynamically trivial. It is therefore natural to ask whether \(\text{\normalfont PL}_q^p({{\mathbf{R}}^n})\) is connected by absolutely continuous arcs in the following sense.

We finally mention that Problem 1 has been solved, in the case \(q=\infty\), in a work in progress by the author [29]. The general case with \(1<q<\infty\) remains open.

Another natural question concerns the stationary points of the minimizing movement scheme of \(\text{\normalfont Isop}\). Let \(\mu\in \text{\normalfont PL}_\infty^\infty(\mathbf{R}^n)\). We say that \(\mu\) is stationary for the isoperimetric minimizing movement if every GMM for \(\operatorname{Isop}\) starting from \(\mu\) is constant. The collection of these probability measures is denoted by \(\text{\normalfont Crit}(\text{\normalfont Isop})\).

Such measures should be regarded as critical points of the isoperimetric functional, in the variational sense given by the minimizing movement scheme. They play the role of configurations for which the discrete implicit Euler scheme detects no descending direction.

A basic class of stationary configurations is given by normalized Lebesgue measures on Euclidean balls – as these are in fact the only global minima of \(\text{\normalfont Isop}\). More generally, one is led to expect that a particular subfamily of weighted sums of uniform measures on pairwise disjoint closed balls are stationary for the isoperimetric minimizing movement.

Finally, we ask for which initial data the minimizing movements for \(\text{\normalfont Isop}\) always converge, as \(t\to\infty\), to a stationary configuration, and for which of these initial data the limiting configuration is uniquely determined by the initial one.

References↩︎