2.1. Notation and convention

• • We denote by $\mathbb {R}^{n \times m}$ the space of $n \times m$ real matrices. If $m = n$ , we simply write it as $\mathbb {M}^{n}$ . Moreover, we use $\mathbb {S}^n$ , $\mathbb {S}_+^n$ and $\mathbb {S}^n_{++}$ to denote symmetric matrices, positive semi-definite matrices and positive definite matrices, respectively. $\mathbb {A}^n$ denotes the space of $n \times n$ antisymmetric matrices.

• • We denote by $|\cdot |$ the Euclidean norm on $\mathbb {R}^n$ . We equip the matrix space $\mathbb {R}^{n \times m}$ with the Frobenius inner product $A \cdot B = Tr(A^{\mathrm {T}} B)$ and the associated norm $\lVert A \rVert _{\mathrm {F}} = \sqrt { A \cdot A}$ .

• • The symmetric and antisymmetric parts of $A \in \mathbb {M}^n$ are given by

  (2.1) \begin{equation} A^{\mathrm {sym}} = (A + A^{\mathrm {T}})/2\,,\quad A^{\mathrm {ant}} = (A - A^{\mathrm {T}})/2\,, \end{equation}
  respectively. We also write $A \preceq B$ (resp., $A \prec B$ ) for $A, B \in \mathbb {S}^n$ if $B - A \in \mathbb {S}^n_+$ (resp., $B - A \in \mathbb {S}^n_{++}$ ).

• • $\mathcal {X}$ denotes a generic compact separable metric space with Borel $\sigma$ -algebra $\mathscr {B}(\mathcal {X})$ , unless otherwise specified.

• • $C(\mathcal {X},\mathbb {R}^n)$ denotes the space of $\mathbb {R}^n$ -valued continuous functions on $\mathcal {X}$ with the supremum norm $\lVert \cdot \rVert _\infty$ . Its dual space, denoted by $\mathcal {M}(\mathcal {X},\mathbb {R}^n)$ , is $\mathbb {R}^n$ -valued Radon measure space with the total variation norm $\lVert \cdot \rVert _{\mathrm {TV}}$ .

• • Let $\mathcal {B}$ be a Banach space with the dual space $\mathcal {B}^*$ . We denote by $\langle \cdot, \cdot \rangle _{\mathcal {B}}$ the duality pairing between $\mathcal {B}$ and $\mathcal {B}^*$ . When $\mathcal {B} = C(\mathcal {X},\mathbb {R}^n)$ , we usually write it as $\langle \cdot, \cdot \rangle _{\mathcal {X}}$ for short. We will also consider the weak and weak^* convergences on $\mathcal {B}$ and $\mathcal {B}^*$ , respectively. In particular, a sequence of measures $\{\mu _j\}$ weak^* converges to $\mu \in \mathcal {M}(\mathcal {X},\mathbb {R}^n)$ if for any $\phi \in C(\mathcal {X},\mathbb {R}^n)$ , there holds $\langle \mu _j, \phi \rangle _{\mathcal {X}} \to \langle \mu, \phi \rangle _{\mathcal {X}}$ as $j \to +\infty$ .

• • Let $\mathbb {R}_+\,:\!=\, [0,\infty )$ , and $\mathcal {M}(\mathcal {X},\mathbb {R}_+)$ be the space of nonnegative finite Radon measures. For $\mu \in \mathcal {M}(\mathcal {X},\mathbb {R}^n)$ , we have an associated variation measure $|\mu |\in \mathcal {M}(\mathcal {X},\mathbb {R}_+)$ such that $\mathrm {d} \mu = \sigma \mathrm {d} |\mu |$ with $|\sigma (x)| = 1$ for $|\mu |$ -a.e. $x \in \mathcal {X}$ , where $\sigma \,:\, \mathcal {X} \to \mathbb {R}^n$ is the Radon–Nikodym derivative (density) of $\mu$ with respect to $|\mu |$ [Reference Evans and Gariepy36, Reference Rudin85].

• • We identify the space of matrix-valued Radon measures $\mathcal {M}(\mathcal {X},\mathbb {R}^{n \times m})$ with $\mathcal {M}(\mathcal {X},\mathbb {R}^{nm})$ by vectorisation. It is easy to see that both sets of $\mathbb {S}^n$ -valued Radon measures $\mathcal {M}(\mathcal {X},\mathbb {S}^n)$ and $\mathbb {S}^n_+$ -valued Radon measures $\mathcal {M}(\mathcal {X},\mathbb {S}_+^n)$ are closed in $\mathcal {M}(\mathcal {X},\mathbb {M}^n)$ with respect to the weak^* topology [Reference Duran and Lopez-Rodriguez35, Theorem 3.5]. Moreover, we have the following characterisation:

  \begin{equation*} (C(\mathcal {X}, \mathbb {S}^n))^* \simeq (C(\mathcal {X},\mathbb {M}^n) /C(\mathcal {X}, \mathbb {A}^n))^* \simeq \mathcal {M}(\mathcal {X}, \mathbb {S}^n)\,, \end{equation*}
  where $\simeq$ means the isometric isomorphism and $C(\mathcal {X},\mathbb {M}^n) /C(\mathcal {X}, \mathbb {A}^n)$ is the quotient space. Indeed, we observe that $\mu \in \mathcal {M}(\mathcal {X},\mathbb {S}^n) \subset \mathcal {M}(\mathcal {X},\mathbb {M}^n) \simeq C(\mathcal {X},\mathbb {M}^n)^*$ if and only if its induced linear functional on $C(\mathcal {X},\mathbb {M}^n)$ has the kernel $C(\mathcal {X}, \mathbb {A}^n)$ , which yields, by [Reference Brezis17, Proposition 11.9],
  \begin{equation*} (C(\mathcal {X},\mathbb {M}^n) /C(\mathcal {X}, \mathbb {A}^n))^* \simeq \mathcal {M}(\mathcal {X}, \mathbb {S}^n)\,. \end{equation*}
  Meanwhile, $C(\mathcal {X}, \mathbb {S}^n) \simeq C(\mathcal {X},\mathbb {M}^n) /C(\mathcal {X}, \mathbb {A}^n)$ is a consequence of $\mathbb {S}^n \perp \mathbb {A}^n$ and $\mathbb {S}^n \simeq \mathbb {M}^n/\mathbb {A}^n$ .

• • For $\mu \in \mathcal {M}(\mathcal {X}, \mathbb {S}_+^n)$ , we define an associated trace measure $Tr\mu$ by the set function $E \to Tr (\mu (E))$ , $E \in \mathscr {B}(\mathcal {X})$ . It is clear that $ 0 \preceq \mu (E) \preceq Tr (\mu (E)) I$ and $ Tr\mu$ is equivalent to $|\mu |$ , denoted by $Tr\mu \sim |\mu |$ . That is,

  (2.2) \begin{equation} |\mu | \ll Tr\mu \quad \text {and} \quad Tr\mu \ll |\mu |\,. \end{equation}
  We will usually use $Tr\mu$ as the dominant measure for $\mu \in \mathcal {M}(\mathcal {X},\mathbb {S}_+^n)$ . In addition, note that for $\lambda \in \mathcal {M}(\mathcal {X}, \mathbb {R}_+)$ with $|\mu | \ll \lambda$ , there holds $\frac {\mathrm {d} \mu }{\mathrm {d} \lambda } \in \mathbb {S}^n_+$ for $\lambda$ -a.e. $x \in \mathcal {X}$ , which is an equivalent characterisation of $\mathcal {M}(\mathcal {X}, \mathbb {S}_+^n)$ .

• • We will use sans serif letterforms to denote vector-valued or matrix-valued measures, e.g., $\mathsf {A} \in \mathcal {M}(\mathcal {X}, \mathbb {M}^n)$ , while letters with serifs are reserved for their densities with respect to some reference measure, e.g., $A_\lambda \,:\!=\, \frac {\mathrm {d} \mathsf {A}}{\mathrm {d} \lambda }$ for $|\mathsf {A}| \ll \lambda$ . The symmetric and antisymmetric parts $\mathsf {A}^{\mathrm {sym}}$ and $\mathsf {A}^{\mathrm {ant}}$ of $\mathsf {A} \in \mathcal {M}(\mathcal {X}, \mathbb {M}^n)$ are defined as in (2.1).

• • We identify a measure and its density with respect to the Lebesgue measure (if exists) unless otherwise specified.

• • For $\lambda \in \mathcal {M}(\mathcal {X}, \mathbb {R}_+)$ , we denote by $L^p_\lambda (\mathcal {X},\mathbb {R}^n)$ with $p \in [1, +\infty ]$ the standard space of $p$ -integrable $\mathbb {R}^n$ -valued functions. For $\mathsf {G} \in \mathcal {M}(\mathcal {X}, \mathbb {S}_+^n)$ , we consider the space of $\mathbb {R}^{n \times m}$ -valued measurable functions endowed with the semi-inner product:

  (2.3) \begin{equation} \langle P, Q \rangle _{L^2_{\mathsf {G}}(\mathcal {X})} \,:\!=\, \langle \mathsf {G}, QP^{\mathrm {T}} \rangle _{\mathcal {X}} = \int _{\mathcal {X}} P \cdot (\mathrm {d} \mathsf {G}\, Q) = \int _{\mathcal {X}} P \cdot \big (G_\lambda Q \big )\, \mathrm {d} \lambda \,, \end{equation}
  where $\lambda$ is a reference measure such that $|\mathsf {G}|\ll \lambda$ and $G_\lambda$ is the density. Noting that $\lVert Q \rVert _{L^2_{\mathsf {G}}(\mathcal {X})} = 0$ is equivalent to $G_\lambda Q = 0$ for $\lambda$ -a.e. $x \in \mathcal {X}$ , the kernel of the seminorm $\lVert \cdot \rVert _{L^2_{\mathsf {G}}(\mathcal {X})}$ is given by $\{Q\,;\ \mathrm {Ran}(Q) \in \mathrm {Ker}(G_\lambda )\,, \,\lambda \text {-a.e.}\}$ . Then, we define the Hilbert space $L^2_{\mathsf {G}}(\mathcal {X}, \mathbb {R}^{n \times m})$ as the quotient space by $\mathrm {Ker}\big (\lVert \cdot \rVert _{L^2_{\mathsf {G}}(\mathcal {X})}\big )$ .

2.2. Preliminaries

We denote by $A^\dagger \in \mathbb {R}^{m \times n}$ the pseudoinverse of a matrix $A \in \mathbb {R}^{n \times m}$ . If $A \in \mathbb {S}^n$ has the eigendecomposition $A = O \Sigma O^{\mathrm {T}}$ , then $A^\dagger = O \Sigma ^\dagger O^{\mathrm {T}}$ with $\Sigma ^\dagger = \text {diag}(\lambda _1^{-1}, \ldots, \lambda _s^{-1},0, \ldots, 0)$ , where $O$ is an orthogonal matrix and $\Sigma = \text {diag}(\lambda _1,\ldots, \lambda _s,0, \ldots, 0)$ is a diagonal matrix with $\{\lambda _i\}$ being nonzero eigenvalues of $A$ .

The next lemma is about the measurability of matrix-valued functions.

The first and second properties are from [Reference Reid81] and [Reference Robertson and Rosenberg82] with the continuity of $A^{1/2}$ in $A \in \mathbb {S}^n_+$ , respectively. In fact, Powers–Størmer inequality [Reference Powers and Størmer80] gives

(2.5) \begin{align} \big \lVert \sqrt {A} - \sqrt {B}\,\big \rVert _{\mathrm {F}}^2 \le \sqrt {n} \lVert A - B \rVert _{\mathrm {F}}\,, \quad \forall A,B \in \mathbb {S}^n_+\,. \end{align}

We finally recall some concepts and useful results from convex analysis. Let $f\,:\,X \to \mathbb {R} \cup \{+ \infty \}$ be an extended real-valued function on a Banach space $X$ . We denote by $\partial f(x)$ its subgradient at $x \in X$ and by $dom(f) \,:\!=\, f^{-1}(\mathbb {R})$ its domain. We say that $f$ is proper if $dom(f) \neq \varnothing$ ; and that $f$ is positively homogeneous of degree $k$ if for all $x \in X$ and $\alpha \gt 0$ , $f(\alpha x) = \alpha ^k f(x)$ . The conjugate function $f^*$ of $f$ is defined by

(2.6) \begin{equation} f^*(x^*) = \sup _{x \in X} \langle x^*, x\rangle _X - f(x)\,, \quad \forall x^* \in X^*\,, \end{equation}

which is convex and lower semicontinuous with respect to the weak^* topology of $X^*$ . The following two lemmas are from [Reference Barbu and Precupanu4, Proposition 2.33] and [Reference Bouchitté12, Proposition 2.5], respectively.

Lemma 2.5 (Fenchel–Rockafellar duality). Let $X$ and $Y$ be two Banach spaces and $L\,:\, X \to Y$ be a bounded linear operator with the adjoint $L^*\,:\, Y^* \to X^*$ . Let $f$ and $g$ be two proper lower semicontinuous convex functions defined on $X$ and $Y$ valued in $\mathbb {R} \cup \{+\infty \}$ , respectively. If there exists $x \in dom (f)$ such that $g$ is continuous at $Lx$ , then

(2.7) \begin{equation} \sup _{x \in X} - f({-}x) - g(Lx) = \inf _{y^* \in Y^*}f^*(L^*y^*) + g^*(y^*)\,, \end{equation}

and the $\inf$ in (2.7) can be attained. Moreover, the $\sup$ in (2.7) is attained at $x \in X$ if and only if there exists a $y^* \in Y^*$ such that $L x \in \partial g^*(y^*)$ and $L^* y^* \in \partial f({-}x)$ , in which case $y^*$ also achieves the $\inf$ in (2.7).

3.1. Action functional

To define our dynamic OT model over the space of $\mathbb {S}^n_+$ -valued measures, the starting point is a weighted action functional. Let $n, k, m \in \mathbb {N}$ be positive integers and $\Lambda \,:\!=\, (\Lambda _1,\Lambda _2)$ be a pair of matrices with $\Lambda _1 \in \mathbb {S}^k_+$ and $\Lambda _2 \in \mathbb {S}_+^m$ . We define the following closed convex set:

(3.1) \begin{align} \mathcal {O}_\Lambda = \Big \{(A,B,C) \in \mathbb {S}^n \times \mathbb {R}^{n \times k} \times \mathbb {R}^{n \times m}\,;\ A + \frac {1}{2} B \Lambda ^2_1 B^{\mathrm {T}} + \frac {1}{2} C \Lambda ^2_2 C^{\mathrm {T}} \preceq 0 \Big \}\,. \end{align}

Note that its characteristic function:

\begin{equation*} \iota _{\mathcal {O}_\Lambda } \,:\!=\, \begin{cases} 0, & (A,B,C) \in \mathcal {O}_\Lambda \,,\\ + \infty, & (A,B,C) \notin \mathcal {O}_\Lambda \,,\\ \end{cases} \end{equation*}

is proper lower semicontinuous and convex [Reference Bauschke and Combettes6, Lemma 1.24]. We denote by $J_\Lambda$ the conjugate function (2.6) of $\iota _{\mathcal {O}_\Lambda }$ and derive the explicit expressions for $J_\Lambda$ and its subgradient $\partial J_\Lambda$ .

Proposition 3.1. $J_\Lambda$ is proper, positively homogeneous of degree one, lower semicontinuous and convex with the following representation:

(3.2) \begin{equation} J_\Lambda (X, Y, Z) = \frac {1}{2} (Y \Lambda _1^\dagger ) \cdot (X^{\dagger } Y \Lambda _1^\dagger ) + \frac {1}{2} (Z \Lambda _2^\dagger ) \cdot (X^{\dagger } Z \Lambda _2^\dagger )\,, \end{equation}

if $X \in \mathbb {S}_+^n$ , $\mathrm {Ran} (Y^{\mathrm {T}}) \subset \mathrm {Ran} ( \Lambda _1)$ , $\mathrm {Ran} (Z^{\mathrm {T}}) \subset \mathrm {Ran} (\Lambda _2)$ and $\mathrm {Ran} ([Y,Z]) \subset \mathrm {Ran}(X)$ ; otherwise, $J_\Lambda (X, Y, Z) = +\infty$ . Moreover, the subgradient of $J_\Lambda$ at $(X,Y, Z) \in dom (J_\Lambda )$ is characterised by

(3.3) \begin{equation} \partial J_\Lambda (X,Y, Z) = \Big \{(A,B,C) \in \mathcal {O}_\Lambda \,; \ Y = X B \Lambda ^2_1\,, \ Z = X C \Lambda ^2_2\,, \ X \cdot \Big (A + \frac {1}{2} B \Lambda _1^2 B^{\mathrm {T}} + \frac {1}{2} C \Lambda ^2_2 C^{\mathrm {T}} \Big ) = 0 \Big \}\,. \end{equation}

$\partial J_\Lambda (X, Y, Z)$ is a singleton if and only if $(X, Y, Z) \in \mathbb {S}^n_{++} \times \mathbb {R}^{n \times k} \times \mathbb {R}^{n \times m}$ and $\Lambda _1 \in \mathbb {S}_{++}^k$ , $\Lambda _2 \in \mathbb {S}_{++}^m$ .

Proof. The properties of $J_\Lambda$ are by [Reference Bauschke and Combettes6, Proposition 14.11]. To derive the formula (3.2), by definition, we have

(3.4) \begin{align} J_\Lambda (X, Y, Z) &= \sup _{(A,B,C) \in \mathcal {O}_\Lambda } X \cdot A + Y \cdot B + Z \cdot C \,, \end{align}

for $(X, Y, Z) \in \mathbb {S}^n \times \mathbb {R}^{n \times k} \times \mathbb {R}^{n \times m}$ . We consider the following four cases.

Case I: $X \in \mathbb {S}^n\backslash \mathbb {S}_+^n$ . We choose a vector $a \in \mathbb {R}^n$ such that $\langle a, X a \rangle \lt 0$ and set $A = - \lambda a a^{\mathrm {T}}\preceq 0$ with $\lambda \gt 0$ , $B = 0$ and $C = 0$ in (3.4). Then it follows that

\begin{equation*} J_\Lambda (X, Y, Z) \ge \sup _{\lambda \gt 0} X \cdot ({-} \lambda a a^{\mathrm {T}}) = + \infty \,. \end{equation*}

Case II: $\mathrm {Ran} (Y^{\mathrm {T}}) \not \subset \mathrm {Ran} (\Lambda _1)$ or $\mathrm {Ran} (Z^{\mathrm {T}}) \not \subset \mathrm {Ran} (\Lambda _2)$ . It suffices to consider the case $\mathrm {Ran} (Y^{\mathrm {T}}) \not \subset \mathrm {Ran} (\Lambda _1)$ , since the same argument applies to the other one. Without loss of generality, we let $Y = [y_1,\ldots, y_n]^{\mathrm {T}}$ with $y_i \in \mathbb {R}^k$ and $y_1 \notin \mathrm {Ran} (\Lambda _1)$ . Thanks to $\Lambda _1 \in \mathbb {S}^k_+$ , $y_1$ has the orthogonal decomposition:

\begin{equation*} y_1 = y_1^{(1)} + y^{(2)}_1 \quad \text {with}\ y^{(1)}_1 \in \mathrm {Ran}(\Lambda _1)\,,\ y^{(2)}_1 \neq 0 \in \mathrm {Ker} (\Lambda _1)\,. \end{equation*}

Taking $A = 0$ , $B = \lambda \big [y_1^{(2)},0\big ]^{\mathrm {T}}$ with $\lambda \in \mathbb {R}$ and $C = 0$ in (3.4), we have

\begin{equation*} J_\Lambda (X, Y, Z) \ge \sup _{\lambda \gt 0} \lambda \big | y_1^{(2)} \big |^2 = + \infty \,. \end{equation*}

Case III: $\mathrm {Ran} ([Y, Z]) \not \subset \mathrm {Ran}(X)$ . It suffices to consider $\mathrm {Ran}( Y ) \not \subset \mathrm {Ran} (X)$ . We take $(A,B,C)$ in (3.4) as:

\begin{align*} A = - \frac {\lambda ^2}{2} (\mathbb {P}_{\mathrm {Ker}(X)} Y \Lambda _1) (\mathbb {P}_{\mathrm {Ker} (X)} Y \Lambda _1)^{\mathrm {T}}\,, \ B = \lambda \mathbb {P}_{\mathrm {Ker}(X)} Y\,,\ C = 0\,, \end{align*}

with $\lambda \gt 0$ , where $\mathbb {P}_{\mathrm {Ker}(X)}\,:\!=\, I - X^\dagger X$ is the orthogonal projection onto $\mathrm {Ker}(X)$ . A direct computation gives

\begin{align*} J_\Lambda (X,Y, Z) &\ge \sup _{(A,B,0) \in \mathcal {O}_\Lambda } X \cdot A + Y \cdot B \\ & \ge \sup _{\lambda \gt 0} - \frac { \lambda ^2}{2} (\mathbb {P}_{\mathrm {Ker}(X)} Y \Lambda _1)\cdot (X \mathbb {P}_{\mathrm {Ker}(X)} Y \Lambda _1) + \lambda Y \cdot ( \mathbb {P}_{\mathrm {Ker}(X)} Y) \\ & \ge \sup _{\lambda \gt 0} \lambda (\mathbb {P}_{\mathrm {Ker}(X)} Y) \cdot (\mathbb {P}_{\mathrm {Ker}(X)} Y) = + \infty \,, \end{align*}

since there holds $( \mathbb {P}_{\mathrm {Ker}(X)} Y) \cdot (\mathbb {P}_{\mathrm {Ker}(X)} Y) \gt 0$ by $\mathrm {Ran}( Y ) \not \subset \mathrm {Ran} (X)$ .

Case IV: $(X, Y,Z) \in \mathbb {S}_+^{n} \times \mathbb {R}^{n \times k} \times \mathbb {R}^{n \times m}$ with $\mathrm {Ran} (Y^{\mathrm {T}}) \subset \mathrm {Ran} (\Lambda _1)$ , $\mathrm {Ran} (Z^{\mathrm {T}}) \subset \mathrm {Ran} (\Lambda _2)$ and $\mathrm {Ran} ([Y, Z]) \subset \mathrm {Ran}(X)$ . For this case, we directly compute

(3.5) \begin{align} X \cdot A + Y \cdot B + Z \cdot C = & X \cdot \Big (A + \frac {1}{2} B \Lambda ^2_1 B^{\mathrm {T}} + \frac {1}{2} C \Lambda ^2_2 C^{\mathrm {T}} \Big ) + Y \cdot B + Z \cdot C - X \cdot \Big ( \frac {1}{2} B \Lambda _1^2 B^{\mathrm {T}} + \frac {1}{2} C \Lambda ^2_2 C^{\mathrm {T}} \Big )\,, \end{align}

and

(3.6) \begin{align} Y \cdot B + Z \cdot C - \frac {1}{2} X \cdot \big (B \Lambda _1^2 B^{\mathrm {T}} + C \Lambda ^2_2 C^{\mathrm {T}} \big ) = & - \frac {1}{2} \Big \lVert \sqrt {X}B \Lambda _1 - \sqrt {X}^{\dagger } Y \Lambda _1^\dagger \Big \lVert _{\mathrm {F}}^2 - \frac {1}{2} \Big \lVert \sqrt {X}C \Lambda _2 - \sqrt {X}^{\dagger } Z \Lambda _2^\dagger \Big \lVert _{\mathrm {F}}^2 \notag \\ & + \frac {1}{2 } \Big \lVert \sqrt {X}^{\dagger } Y \Lambda _1^\dagger \Big \lVert _{\mathrm {F}}^2 + \frac {1}{2} \Big \lVert \sqrt {X}^{\dagger } Z \Lambda _2^\dagger \Big \lVert _{\mathrm {F}}^2\,, \end{align}

where we have used

\begin{equation*} Y \cdot B + Z \cdot C = \big (\sqrt {X} \sqrt {X}^\dagger Y \Lambda _1^\dagger \Lambda _1 \big ) \cdot B + \big (\sqrt {X} \sqrt {X}^\dagger Z \Lambda _2^\dagger \Lambda _2 \big ) \cdot C \,,\end{equation*}

by the range relations: $\mathrm {Ran} (Y^{\mathrm {T}}) \subset \mathrm {Ran} (\Lambda _1)$ , $\mathrm {Ran} (Z^{\mathrm {T}}) \subset \mathrm {Ran} (\Lambda _2)$ and $\mathrm {Ran} ([Y,Z]) \subset \mathrm {Ran}(X)$ . Also, by (3.1), we have $ X \cdot \big (A + \frac {1}{2} B \Lambda _1^2 B^{\mathrm {T}} + \frac {1}{2} C \Lambda _2^2 C^{\mathrm {T}} \big ) \le 0$ . Hence, by (3.5) and (3.6), the maximisers to (3.4) are given by the set

(3.7) \begin{equation} \Big \{(A,B,C) \in \mathcal {O}_\Lambda \,; \ Y = X B \Lambda ^2_1 \,, \ Z = X C \Lambda ^2_2\,, \ X \cdot \Big (A + \frac {1}{2} B \Lambda _1^2 B^{\mathrm {T}} + \frac {1}{2} C \Lambda ^2_2 C^{\mathrm {T}} \Big ) = 0 \Big \}\,, \end{equation}

and the corresponding supremum is (3.2).

Finally, to characterise the subgradient of $J_\Lambda$ , by Lemma2.4, we have that $(A,B,C) \in \partial J_\Lambda (X,Y,Z)$ if and only if $(A,B,C) \in \mathcal {O}_\Lambda$ and $ J_\Lambda (X,Y,Z) = X \cdot A + Y \cdot B + Z \cdot C$ holds. Then, (3.3) readily follows from the above argument. For the last statement, we note that $\partial J_\Lambda (X,Y,Z)$ is a singleton if and only if the equations in (3.3) for $(A,B,C)$ are uniquely solvable, which is equivalent to $\Lambda _1 \in \mathbb {S}_{++}^k$ , $\Lambda _2 \in \mathbb {S}_{++}^m$ and $X \in \mathbb {S}_{++}^n$ .

Similarly to the unbalanced WFR distance [Reference Chizat, Peyré, Schmitzer and Vialard27, Reference Kondratyev, Monsaingeon and Vorotnikov56, Reference Liero, Mielke and Savaré64], the variables $(X, Y, Z) \in \mathbb {S}^n \times \mathbb {R}^{n \times k} \times \mathbb {R}^{n \times m}$ in the infinitesimal cost $J_\Lambda (X, Y, Z)$ represent the mass, the momentum for the mass transportation and the source for the mass variation, respectively, in our transport problem (see Remark3.6 and Definition3.8). In what follows, we assume $m = n$ , since the dimensions of the mass $X \in \mathbb {S}^n$ and the source $Z \in \mathbb {R}^{n \times m}$ need to match. We shall also let $\Lambda _2 \in \mathbb {S}^n_{++}$ to avoid technical issues (see Remark3.10). Now, for a given triplet of measures $\mu \,:\!=\, \mathsf {(G,q, R)} \in \mathcal {M}(\mathcal {X}, \mathbb {S}^n \times \mathbb {R}^{n \times k} \times \mathbb {M}^n)$ , we define a positive measure $\mathcal {J}_{\Lambda } (\mu )$ on $\mathcal {X}$ by

(3.8) \begin{equation} \mathcal {J}_{\Lambda }(\mu )(E)\,:\!=\, \int _E J_\Lambda \left (\frac {\mathrm {d} \mu }{\mathrm {d} \lambda }\right ) \mathrm {d} \lambda \,, \end{equation}

for a measurable set $E \in \mathscr {B}(\mathcal {X})$ , where $\lambda \in \mathcal {M}(\mathcal {X},\mathbb {R}_+)$ is a reference measure such that $|\mu | \ll \lambda$ . Thanks to the positive homogeneity of $J_\Lambda$ by Proposition3.1, the definition (3.8) of $\mathcal {J}_{\Lambda }$ is independent of the choice of $\lambda$ . To alleviate notations, we adopt the following conventions in the rest of this work.

• • We define the space $\mathbb {X} \,:\!=\, \mathbb {S}^n \times \mathbb {R}^{n \times k} \times \mathbb {M}^n$ and then write $\mathcal {M}(\mathcal {X},\mathbb {X}) = \mathcal {M}(\mathcal {X}, \mathbb {S}^n \times \mathbb {R}^{n \times k} \times \mathbb {M}^n) = C(\mathcal {X},\mathbb {X})^*$ , where $C(\mathcal {X},\mathbb {X}) = C(\mathcal {X}, \mathbb {S}^n \times \mathbb {R}^{n \times k} \times \mathbb {M}^n)$ .

• • We often write $\mu$ for $\mathsf {(G,q,R)} \in \mathcal {M}(\mathcal {X},\mathbb {X})$ for short, which will be clear from the context.

• • We write $\mathcal {J}_{\Lambda }(\mu )(E)$ as $\mathcal {J}_{\Lambda, E}(\mu )$ for short. Then, $\mathcal {J}_{\Lambda, \mathcal {X}}(\mu )$ denotes the total measure $\mathcal {J}_{\Lambda }(\mu )(\mathcal {X})$ .

• • We denote by $(G_\lambda, q_\lambda, R_\lambda )$ the density of $\mathsf {(G,q,R)} \in \mathcal {M}(\mathcal {X},\mathbb {X})$ with respect to a reference measure $\lambda \in \mathcal {M}(\mathcal {X},\mathbb {R}_+)$ such that $|\mathsf {(G,q,R)}| \ll \lambda$ . The subscript $\lambda$ of $(G_\lambda, q_\lambda, R_\lambda )$ will often be omitted for simplicity.

• • The generic positive constant $C$ involved in the estimates below may change from line to line.

By Proposition3.1 and the formula (3.8), we have the following useful lemma.

Lemma 3.3. For $\mu = \mathsf {(G,q,R)} \in \mathcal {M}(\mathcal {X},\mathbb {X})$ with $\mathcal {J}_{\Lambda, \mathcal {X}}(\mu ) \lt + \infty$ , we have $\mathsf {G} \in \mathcal {M}(\mathcal {X},\mathbb {S}_+^n)$ and $|(\mathsf {q}, \mathsf {R})| \ll Tr \mathsf {G}$ with

(3.9) \begin{equation} G_\lambda \in \mathbb {S}_+^n,\ \mathrm {Ran}\left ([q_\lambda, R_\lambda ]\right ) \subset \mathrm {Ran} \left (G_\lambda \right )\!, \ \mathrm {Ran}(q_\lambda ^{\mathrm {T}}) \subset \mathrm {Ran}( \Lambda _1), \ \mathrm {Ran}(R_\lambda ^{\mathrm {T}}) \subset \mathrm {Ran}(\Lambda _2),\quad \text {$\lambda $-a.e.}\,. \end{equation}

Proof. By $\mathcal {J}_{\Lambda, \mathcal {X}}(\mu ) = \int _{\mathcal {X}} J_\Lambda (\mu _\lambda )\, \mathrm {d} \lambda \lt + \infty$ , $J_\Lambda (\mu _\lambda )$ is finite for $\lambda$ -a.e. $x \in \mathcal {X}$ , where $\mu _\lambda = (G_\lambda, q_\lambda, R_\lambda )$ . It means that $\mu _\lambda (x) \in dom(J_{\Lambda })$ holds $\lambda$ -a.e., which immediately gives (3.9) by Proposition3.1. We next show the absolute continuity of $|\mathsf {q}|$ and $|\mathsf {R}|$ with respect to $Tr \mathsf {G}$ , that is, for $E \in \mathscr {B}(\mathcal {X})$ with $Tr \mathsf {G} (E) = 0$ , we have $|\mathsf {q}|(E) = |\mathsf {R}|(E) = 0$ . For this, we consider two measurable subsets $E_1$ and $E_2$ of $E$ with $E = E_1 \cup E_2$ :

\begin{equation*} E_1 = \left \{x \in E\,;\ G_\lambda (x) \in \mathbb {S}_+^n\backslash \{0\}\right \},\quad E_2 = \left \{x \in E\,;\ G_\lambda (x) = 0\right \}. \end{equation*}

By $Tr \mathsf {G}(E_1) = 0$ and $Tr G_\lambda \gt 0$ on $E_1$ everywhere, we have $\lambda (E_1) = 0$ . Then $|\mathsf {q}|(E_1) = 0$ and $|\mathsf {R}|(E_1) = 0$ follows from $|\mathsf {q}|, |\mathsf {R}| \ll \lambda$ . Moreover, by (3.9) and $G_\lambda = 0$ on $E_2$ , we have $q_\lambda (x) = 0$ and $R_\lambda (x) = 0$ for $\lambda$ -a.e. $x \in E_2$ . Then it follows that $|\mathsf {q}|(E_2) = 0$ and $|\mathsf {R}|(E_2) = 0$ . The proof is complete.

3.2. Continuity equation

Another key ingredient for the dynamic OT formulation is a matricial continuity equation; see Definition3.4 below. Let us fix more notations.

• • Let $\Omega \subset \mathbb {R}^d$ be a compact set with a nonempty interior, a smooth boundary $\partial \Omega$ and the exterior unit normal vector $\nu = (\nu _1,\ldots, \nu _d)$ . We denote by $Q_a^b \,:\!=\, [a,b] \times \Omega \subset \mathbb {R}^{1 + d}$ with $b \gt a \gt 0$ the associated time-space domain. If $[a,b] = [0,1]$ , we simply write it as $Q$ .

• • For a function $\Phi (t,x)$ on $Q_a^b$ , we write $\Phi _t(\cdot ) \,:\!=\, \Phi (t,\cdot )$ if we regard it as a family of functions $\{\Phi _t\}_{t \in [a,b]}$ in $x$ .

• • We denote by $\pi ^t\,:\, (t,x) \to t$ the projection. We use the subscript $\#$ to denote the pushforward by a map. For instance, for a measure $\mu$ on $Q_a^b$ , $\pi ^t_\# \mu = \mu \circ (\pi ^t)^{-1}$ is the pushforward measure on $[a,b]$ .

• • Let $X$ and $Y$ be two Banach spaces. We denote by $\mathcal {L}(X,Y)$ the space of continuous linear operators from $X$ to $Y$ (simply $\mathcal {L}(X)$ if $X = Y$ ) and by $C_c^\infty (\mathbb {R}^d,X)$ the $X$ -valued smooth functions with compact support. We also need $C^k$ -smooth functions $C^k(\Omega, X)$ , where we assume that the derivatives exist in the interior of $\Omega$ and can be continuously extended to the boundary. The norm on $C^k(\Omega, X)$ is defined by $\lVert \Phi \rVert _{k,\infty } \,:\!=\, \sum _{|\alpha | \le k} \sup _{x \in \Omega } \lVert D^\alpha \Phi (x) \rVert$ . Other similar notations are interpreted accordingly.

• • We recall the indicator function of a set $A$ :

  (3.10) \begin{equation} \chi _A(x) = \begin{cases} 1, & \text {if}\ x \in A\,,\\ 0, & \text {if}\ x \notin A\,. \end{cases} \end{equation}

• • We use $\widehat{\cdot}$ to denote the Fourier transform of a function, or the symbol of a constant coefficient linear differential operator.

Let $\mathsf {D}^*\,:\,C_c^\infty (\mathbb {R}^d, \mathbb {S}^n) \to C_c^\infty (\mathbb {R}^d, \mathbb {R}^{n \times k})$ be a general first-order constant coefficient linear differential operator satisfying $\mathsf {D}^*(I) = 0$ . That is, for a matrix-valued function $\Phi \in C_c^\infty (\mathbb {R}^d, \mathbb {S}^n)$ with components $\{\Phi _{ij}\}_{i,j = 1}^n$ , we have

(3.11) \begin{equation} \mathsf {D}^*(\Phi _{ij}(e_{ij} + e_{ji})) = A_0^{ij}\Phi _{ij}(x) + \sum _{l = 1}^d A_l^{ij} \partial _{x_l} \Phi _{ij}(x)\,, \quad i \le j\,, \end{equation}

for some matrices $\{A_l^{ij}\}_{l = 0}^d \subset \mathbb {R}^{n \times k}$ , and there holds $\sum _{i = 1}^n A_0^{ii} = 0$ . Here $e_{ij}$ is the $n \times n$ matrix unit with $1$ at the $(i,j)$ -entry. By Fourier transform, the operator $\mathsf {D}^*$ can be equivalently characterised by

(3.12) \begin{equation} \mathsf {D}^* (\Phi )(x) = \int _{\mathbb {R}^d} \widehat {\mathsf {D}^*}(\xi )\big [\widehat {\Phi }(\xi )\big ] e^{\mathrm {i} \xi \cdot x}\, \mathrm {d} \xi \,, \quad \Phi \in C_c^\infty (\mathbb {R}^d, \mathbb {S}^n)\,, \end{equation}

where $\widehat {\Phi }(\xi )$ is the Fourier transform of $\Phi$ :

\begin{equation*} \widehat {\Phi }(\xi ) = \frac {1}{(2\pi )^d} \int _{\mathbb {R}^d} \Phi (x) e^{-\mathrm {i}\xi \cdot x} \,\mathrm {d} x\,, \end{equation*}

and $\widehat {\mathsf {D}^*}(\xi )\,:\, \mathbb {R}^d \to \mathcal {L}(\mathbb {S}^n,\mathbb {R}^{n \times k})$ is the symbol of $\mathsf {D}^*$ such that for any $X \in \mathbb {S}^n$ and $Y \in \mathbb {R}^{n \times k}$ , $Y \cdot \widehat {\mathsf {D}^*}(\xi )[X]$ is a first-order polynomial in $\xi$ . We write $\widehat {\mathsf {D}^*}(\xi )$ as the sum of its homogeneous components: $\widehat {\mathsf {D}^*}(\xi ) = \widehat {\mathsf {D}^*_0} + \widehat {\mathsf {D}^*_1}(\xi )$ , where $\widehat {\mathsf {D}^*_0}$ and $\widehat {\mathsf {D}^*_1}(\xi )$ are homogeneous of degree $0$ and $1$ , respectively: for $X = (X_{ij}) \in \mathbb {S}^n$ ,

\begin{equation*} \widehat {\mathsf {D}^*_0}[X] = \frac {1}{2} \sum _{i = 1}^n A_0^{ii} X_{ii} + \sum _{i \lt j} A_0^{ij} X_{ij}\,, \end{equation*}

and

\begin{equation*} \widehat {\mathsf {D}^*_1}(\xi )[X] = \frac {\mathrm {i}}{2} \sum _{l = 1}^d\sum _{i = 1}^n A_l^{ii} \xi _l X_{ii} + \mathrm {i} \sum _{l = 1}^d\sum _{i \lt j} A_l^{ij} \xi _l X_{ij}\,, \end{equation*}

with matrices $A_{l}^{ij}$ given in (3.11). Then, noting that the Fourier transform of $I$ is $\delta _0 I$ , it is easy to see that the condition $\mathsf {D}^*(I) = 0$ is equivalent to $\widehat {\mathsf {D}^*}(0)(I) = \widehat {\mathsf {D}_0^*}(I) = \frac {1}{2} \sum _{i = 1}^n A_0^{ii} = 0$ .

By abuse of notation, we define $\mathsf {D}^*\Phi$ for functions $\Phi (t,x)$ on $\mathbb {R}^{1 + d}$ by acting $\mathsf {D}^*$ on the spatial variable $x$ . Moreover, we define the operator $\mathsf {D}$ as the adjoint operator of $ - \mathsf {D}^*$ in the sense of distribution, which can be viewed as a bdivergence operator that maps the momentum to the mass (see equation (3.14)). We similarly denote by $\mathsf {D}_0$ and $\mathsf {D}_1$ the homogeneous parts of degree $0$ and $1$ of the operator $\mathsf {D}$ , respectively.

Example 3.1. A simple example of $\mathsf {D}$ is the entry-wise transport, in which case the mass transportation between components is forbidden. To be precise, for $\mathsf {q} \in \mathcal {M}(Q, \mathbb {R}^{n \times n \times d})$ , we regard $\mathsf {q}$ as a collection of $\mathbb {R}^d$ -valued measures $\{\mathsf {q}_{ij}\}_{i,j =1}^n \subset \mathcal {M}(Q, \mathbb {R}^{d})$ , and define

\begin{equation*} \mathsf {D}(\mathsf {q}) = (\mathrm{div} \mathsf {q})^{\mathrm {sym}} = \frac {\mathrm {div} \mathsf {q} + (\mathrm {div} \mathsf {q})^{\mathrm {T}}}{2}\,, \end{equation*}

where the standard divergence is applied to each $q_{ij}$ , i.e., $(\mathrm {div} \mathsf {q})_{ij} \,:\!=\, \mathrm {div} q_{ij}$ . Then, the adjoint $\mathsf {D}^*$ is simply given by the gradient that acts on $\Phi \in C_c^\infty (\mathbb {R}^d, \mathbb {S}^n)$ component-wisely: $\mathsf {D}^* \Phi = (\nabla \Phi _{ij})_{ij}$ . More examples with discussion can be found in Section 7 .

Definition 3.4. A measure $\mathsf{G} \in \mathcal {M}(Q_a^b, \mathbb {S}^n)$ connects $\mathsf {G}_a, \mathsf {G}_b \in \mathcal {M}(\Omega, \mathbb {S}^n_+)$ over the time interval $[a,b]$ , if there exists $\mathsf {(q, R)} \in \mathcal {M}(Q_a^b, \mathbb {R}^{n \times k} \times \mathbb {M}^n)$ satisfying the following general matrix-valued continuity equation:

(3.13) \begin{equation} \int _{Q_a^b} \partial _t\Phi \cdot \mathrm {d} \mathsf {G} + \mathsf {D}^* \Phi \cdot \mathrm {d} \mathsf {q} + \Phi \cdot \mathrm {d} \mathsf {R} = \int _{\Omega } \Phi _b \cdot \mathrm {d} \mathsf {G}_b - \int _{\Omega } \Phi _a \cdot \mathrm {d} \mathsf {G}_a\,,\quad \forall \Phi \in C^1(Q_a^b,\mathbb {S}^n)\,. \end{equation}

The measures $\mathsf {G}_a$ and $\mathsf{G}_b$ are referred to as the initial and final distributions of $\mathsf {G}$ , respectively. Moreover, we denote by $\mathcal {CE}([a,b];\,\mathsf {G}_a, \mathsf {G}_b)$ the set of the measures $\mathsf {(G,q, R)} \in \mathcal {M}(Q_a^b,\mathbb {X})$ satisfying (3.13).

Remark 3.5. It is easy to derive the distributional equation of (3.13):

(3.14) \begin{equation} \partial _t \mathsf {G} + \mathsf {D}\mathsf {q} = \mathsf {R}^{\mathrm {sym}}\,, \end{equation}

with the measure $\mathsf {q}$ satisfying a homogeneous boundary condition on $\partial \Omega$ . Indeed, assume that $\mathsf {q}$ admits a smooth density $q$ with respect to the Lebesgue measure. Note that for $\mathsf {D}^* = a + \partial _{x_i}$ with $\mathsf {D} = - a + \partial _{x_i}$ ( $a \in \mathbb {R}$ ), a direct integration by parts gives, for smooth real functions $f,g$ on $\Omega$ ,

\begin{equation*} \int _{\Omega } ((a + \partial _{x_i})f(x)) g(x) + f(x) ({-}a + \partial _{x_i}) g(x) \, \mathrm {d} x = \int _{\partial \Omega } \nu _i f(x) g(x) \, \mathrm {d} x\,. \end{equation*}

We then have, by linearity and noting $\widehat {\partial _{x_k}} = \mathrm {i} \xi _k$ , for a general $\mathsf {D}^*$ ,

\begin{equation*} \int _\Omega \mathsf {D} q \cdot \Phi + q \cdot \mathsf {D}^* \Phi \, \mathrm {d} x = \int _{\partial \Omega } q \cdot \widehat {\mathsf {D}_1^*}({-}\mathrm {i} \nu )(\Phi ) \, \mathrm {d} x = \int _{\partial \Omega } \widehat {\mathsf {D}_1}({-}\mathrm {i} \nu )(q) \cdot \Phi \, \mathrm {d} x\,, \quad \forall \Phi \in C^1(\Omega, \mathbb {S}^n)\,. \end{equation*}

It follows that the boundary condition $\widehat {\mathsf {D}_1}({-}\mathrm {i} \nu )(q) = 0$ holds for $\mathsf {q}$ satisfying (3.13). In the case of $\mathsf {D} = \mathrm {div}$ for $\mathsf {q} \in \mathcal {M}(Q, \mathbb {R}^{d})$ , we see that $\widehat {\mathsf {D}_1}({-}\mathrm {i} \nu )(q) = 0$ is the familiar no-flux boundary condition $\nu \cdot q = 0$ .

The following elementary lemma gives the absolute continuity of the time marginal of $\mathsf {G}$ .

Proof. It suffices to consider $[a,b] = [0,1]$ . By (3.13) with test functions $\Phi (t,x) = \phi (t) \in C_c^1((0,1),\mathbb {S}^n)$ , we have

(3.15) \begin{equation} \int _0^1 \partial _t \phi \cdot \mathrm {d} \pi ^t_\# \mathsf {G} + \phi \cdot \mathrm {d} \pi _\#^t \mathsf {R} = 0\,, \end{equation}

which implies that $(\pi _\#^t \mathsf {R})^{\mathrm {sym}}$ is the distributional derivative of $\pi _\#^t \mathsf {G}$ . Note that $\pi ^t_\# \mathsf {G}$ and $\pi ^t_\# \mathsf {R}$ are Radon measures (since every finite Borel measure on $[0,1]$ is regular). There exists a matrix-valued bounded variation function $M(t)$ that generates the Radon measure $\pi ^t_\# \mathsf {R}$ [Reference Gerald42, Theorem 3.29]. It follows from (3.15) that

(3.16) \begin{equation} \mathrm {d} \pi ^t_\# \mathsf {G} = (M(t)^{\mathrm {sym}} + C)\, \mathrm {d} t\,, \end{equation}

for some $C \in \mathbb {S}^n$ [Reference Gerald42, Theorem 3.36]. bIf $\mathsf {G} \in \mathcal {M}(Q_a^b,\mathbb {S}_+^n)$ , then (3.16) and (2.2) readily give $Tr \pi ^t_\# \mathsf {G} \sim |\pi ^t_\# \mathsf {G}| \ll \mathrm {d} t$ , which further yields $\pi ^t_\# |\mathsf {G}| \ll \mathrm {d} t$ by noting $Tr \pi ^t_\# \mathsf {G} = \pi ^t_\# Tr \mathsf {G} \sim \pi ^t_\# |\mathsf {G}|$ .

3.3. Weighted Wasserstein–Bures distance

We are now ready to define a class of distances on $\mathcal {M}(\Omega, \mathbb {S}^n_+)$ by minimising the action functional $\mathcal {J}_{\Lambda, Q}(\mu )$ over the solutions to the continuity equation (3.13).

Definition 3.8. The weighted Wasserstein–Bures distance between $\mathsf {G}_0, \mathsf {G}_1 \in \mathcal {M}(\Omega, \mathbb {S}_+^n)$ is defined by

(𝒫) \begin{equation} \mathrm {WB}^2_{\Lambda }(\mathsf {G}_0,\mathsf {G}_1) = \inf _{\mu \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)} \mathcal {J}_{\Lambda, Q}(\mu ). \end{equation}

We remark that the quantity $\mathcal {J}_{\Lambda, Q}(\mu )$ can be understood as the energy of the measure $\mu \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ . The following a priori estimate shows that $\mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ is nonempty and $\rm {WB}_{\Lambda }(\mathsf {G}_0,\mathsf {G}_1)$ is always finite, which means that the problem ( $\mathcal {P}$ ) is well defined.

Lemma 3.9. Given $\mathsf {G}_0,\mathsf {G}_1 \in \mathcal {M}(\Omega, \mathbb {S}^n_+)$ , let $\lambda \in \mathcal {M}(\Omega, \mathbb {R}_+)$ be a reference measure such that $|\mathsf {G}_0|, |\mathsf {G}_1| \ll \lambda$ . Then there exists $\mu = \mathsf {(G,0,R)} \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ with finite $\mathcal {J}_{\Lambda, Q}(\mu )$ . Moreover, it holds that

(3.17) \begin{equation} \mathrm {WB}^2_{\Lambda }(\mathsf {G}_0,\mathsf {G}_1) \le \mathrm {WB}^2_{(0,\Lambda _2)}(\mathsf {G}_0,\mathsf {G}_1) \le 2 \big \lVert \Lambda _2^{-1} \big \lVert _{\mathrm {F}}^2 \int _{\Omega } \big \lVert \sqrt {G_{1,\lambda }} - \sqrt {G_{0,\lambda }} \big \lVert _{\mathrm {F}}^2 \ \mathrm {d} \lambda \,, \end{equation}

where $G_{0,\lambda }$ and $G_{1,\lambda }$ are densities of $\mathsf {G}_0$ and $\mathsf {G}_1$ with respect to $\lambda$ .

Proof. We omit the subscript $\lambda$ of $G_{0,\lambda }$ and $G_{1,\lambda }$ for simplicity. We define measures

\begin{equation*} \mathsf {G} \,:\!=\, \left (\sqrt {G_{0}} + t \Big (\sqrt {G_{1}} - \sqrt {G_{0}} \Big ) \right )^2 \mathrm {d} t \otimes \lambda \in \mathcal {M}(Q, \mathbb {S}^n_+)\,, \end{equation*}

and

\begin{equation*} \mathsf {R} \,:\!=\, 2 \left (\sqrt {G_{0}} + t \left (\sqrt {G_{1}} - \sqrt {G_{0}} \right ) \right ) \left (\sqrt {G_{1}} - \sqrt {G_{0}}\right ) \mathrm {d} t \otimes \lambda \subset \mathcal {M}(Q, \mathbb {M}^n)\,, \end{equation*}

which satisfies $\mu = \mathsf {(G,0,R)} \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ and $\mathrm {Ran} \big ( \frac {\mathrm {d} \mathsf {R}}{\mathrm {d} t \otimes \lambda } \big ) \subset \mathrm {Ran} \big ( \frac {\mathrm {d} \mathsf {G}}{\mathrm {d} t \otimes \lambda } \big )$ for $\mathrm {d} t \otimes \lambda$ -a.e. Moreover, we note

\begin{align*} \mathrm {Ran}\left (\sqrt {G_1} - \sqrt {G_0}\right )\subset \mathrm {Ran}\left (\sqrt {G_0} + t \left (\sqrt {G_1} - \sqrt {G_0}\right )\right )\,,\quad t \in (0,1)\,, \end{align*}

from the relation: $ \mathrm {Ker}\big (\sqrt {G_0} + t (\sqrt {G_1} - \sqrt {G_0})\big ) = \mathrm {Ker} \big (\sqrt {G_0}\big ) \cap \mathrm {Ker} \big (\sqrt {G_1}\big )\subset \mathrm {Ker}\big (\sqrt {G_1} - \sqrt {G_0}\big )$ . Then, we compute

(3.18) \begin{equation} \mathcal {J}_{\Lambda, Q}(\mu ) = 2 \int _{\Omega } \Big \lVert \Big (\sqrt {G_1} - \sqrt {G_0}\Big ) \Lambda _2^{-1} \Big \lVert _{\mathrm {F}}^2\ \mathrm {d} \lambda \,, \end{equation}

for $\mu$ defined above. The proof is completed by the submultiplicativity of the Frobenius norm.

Remark 3.11. $\mathrm {WB}_{(0,\Lambda _2)}$ is the matricial Hellinger distance $d_H$ in [Reference Monsaingeon and Vorotnikov73 , Definition 4.1], up to a transformation. Indeed, recalling Lemma 3.3 , we have that if $\Lambda _1 = 0$ , then $\mathsf {q}$ must be zero and ( $\mathcal {P}$ ) reduces to

(3.19) \begin{equation} \mathrm {WB}_{(0,\Lambda _2)}^2(\mathsf {G}_0,\mathsf {G}_1) = \inf \{\mathcal {J}_{(0,\Lambda _2),Q}(\mu )\,;\ \mu = \mathsf {(G,0,R)} \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)\}. \end{equation}

For a given $S \in \mathbb {S}^n_{++}$ , we introduce a linear map $g_{S}(A) \,:\!=\, S A S : \mathbb {S}^n_{+} \to \mathbb {S}^n_{+}$ with the inverse $g_{S^{-1}}$ . It is easy to see that $\mathsf {(G,0,R)} \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ if and only if $(g_{\Lambda _2^{-1}}(\mathsf {G}),0, g_{\Lambda _2^{-1}}(\mathsf {R})) \in \mathcal {CE}([0,1];\,g_{\Lambda _2^{-1}}(\mathsf {G}_0),g_{\Lambda _2^{-1}}(\mathsf {G}_1))$ , and there holds $\mathcal {J}_{(0,\Lambda _2),Q}(\mathsf {(G,0,R)}) = \mathcal {J}_{(0,I),Q}(g_{\Lambda _2^{-1}}(\mathsf {G}),0, g_{\Lambda _2^{-1}}(\mathsf {R}))$ . Therefore, we have

\begin{equation*} \mathrm {WB}_{(0,\Lambda _2)}(\mathsf {G}_0,\mathsf {G}_1) = \mathrm {WB}_{(0,I)}(g_{\Lambda _2^{-1}}(\mathsf {G}_0),g_{\Lambda _2^{-1}}(\mathsf {G}_1))\,. \end{equation*}

From [Reference Monsaingeon and Vorotnikov73 , Definition 4.1] and Theorem 4.5 below, one can see that $\mathrm {WB}_{(0,I)}$ is nothing else than the convex formulation of the Hellinger distance $d_H$ , up to a constant. We refer the readers to [Reference Monsaingeon and Vorotnikov73 , Lemma 4.3 and Theorem 2] for the properties of the Hellinger distance and its relation with the Bures-Wasserstein distance on $\mathbb {S}^n_+$ [10].

3.4. A priori estimate

Thanks to Lemma3.9, the optimisation (𝒫) can be equivalently taken over the following set:

\begin{equation*} \mathcal {CE}_\infty ([0,1];\,\mathsf {G}_0,\mathsf {G}_1)\,:\!=\, \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1) \bigcap \{\mu \in \mathcal {M}(Q,\mathbb {X});\, \mathcal {J}_{\Lambda, Q}(\mu ) \lt +\infty \}\,. \end{equation*}

Before we proceed, we give some auxiliary results. First, we introduce

(3.20) \begin{equation} \mathcal {J}_{\Lambda, \mathcal {X}}^*(\mathsf {G},u,W) \,:\!=\, \frac {1}{2} \lVert (u \Lambda _1, W \Lambda _2) \rVert ^2_{L^2_{\mathsf {G}}(\mathcal {X})}\quad \text {on}\ \mathcal {M}(\mathcal {X},\mathbb {S}_+^n) \times C(\mathcal {X}, \mathbb {R}^{n \times k} \times \mathbb {M}^n)\,, \end{equation}

where $\lVert \cdot \rVert _{L^2_{\mathsf {G}}(\mathcal {X})}$ is defined by (2.3). By an argument similar to the one for Lemma4.1 below, we have that the conjugate function (2.6) of $\mathcal {J}^*_{\Lambda, \mathcal {X}}(\mathsf {G},u,W)$ with respect to $(u,W)$ is exactly $\mathcal {J}_{\Lambda, \mathcal {X}}(\mathsf {G},\mathsf {q},\mathsf {R})$ . Moreover, there holds

(3.21) \begin{equation} \mathcal {J}_{\Lambda, \mathcal {X}}(\mathsf {G},\mathsf {q},\mathsf {R}) = \sup _{(u,W) \in L^\infty _{|\mathsf {(G,q,R)}|}(\mathcal {X}, \mathbb {R}^{n \times k} \times \mathbb {M}^n) } \langle (\mathsf {q},\mathsf {R}), (u,W) \rangle _{\mathcal {X}} - \mathcal {J}_{\Lambda, \mathcal {X}}^*(\mathsf {G},u,W)\,. \end{equation}

Since $\mathcal {J}_{\Lambda, \mathcal {X}}(\mathsf {G}, \mathsf {q},\mathsf {R})$ and $\mathcal {J}_{\Lambda, \mathcal {X}}^*(\mathsf {G}, u, W)$ are homogeneous of degree $2$ in $(\mathsf {q},\mathsf {R})$ and $(u,W)$ , respectively, by (3.21), it holds that for $\mathsf {(G,q,R)} \in \mathcal {M}(\mathcal {X},\mathbb {X})$ and $(u,W) \in L^\infty _{|\mathsf {(G,q,R)}|}(\mathcal {X}, \mathbb {R}^{n \times k} \times \mathbb {M}^n)$ ,

(3.22) \begin{align} \langle (\mathsf {q},\mathsf {R}), (u,W)\rangle _{\mathcal {X}} \le \gamma ^{-2} \mathcal {J}_{\Lambda, \mathcal {X}}(\mathsf {G}, \mathsf {q}, \mathsf {R}) + \gamma ^2\mathcal {J}_{\Lambda, \mathcal {X}}^*(\mathsf {G},u, W)\,, \quad \forall \gamma \gt 0\,. \end{align}

We minimise the right-hand side of (3.22) with respect to $\gamma$ and obtain

(3.23) \begin{align} \langle (\mathsf {q},\mathsf {R}), (u,W)\rangle _{\mathcal {X}} \le 2 \sqrt {\mathcal {J}_{\Lambda, \mathcal {X}}(\mathsf {G}, \mathsf {q}, \mathsf {R}) \mathcal {J}_{\Lambda, \mathcal {X}}^*(\mathsf {G},u, W)}\,, \end{align}

where we have used non-negativity of $\mathcal {J}_{\Lambda, \mathcal {X}}$ and $\mathcal {J}_{\Lambda, \mathcal {X}}^*$ .

Second, we observe from formulas (3.2) and (3.8) and Lemmas2.3 and 3.3 that for $\mu = (\mathsf {G},\mathsf {q},\mathsf {R}) \in \mathcal {M}(\mathcal {X},\mathbb {X})$ with $\mathcal {J}_{\Lambda, \mathcal {X}}(\mu ) \lt +\infty$ , the functions $G_\lambda ^\dagger q_\lambda \Lambda _1^\dagger$ and $G_\lambda ^\dagger R_\lambda \Lambda _2^{-1}$ are well defined, Borel measurable and independent of the reference measure $\lambda$ (hence we omit the subscript $\lambda$ in the sequel for simplicity), and there holds

(3.24) \begin{align} \mathcal {J}_{\Lambda, \mathcal {X}}(\mu ) = \frac {1}{2} \lVert G^\dagger q \Lambda _1^\dagger \rVert ^2_{L^2_{\mathsf {G}}(\mathcal {X})} + \frac {1}{2} \lVert G^\dagger R \Lambda _2^{-1} \rVert ^2_{L^2_{\mathsf {G}}(\mathcal {X})} \lt +\infty \,. \end{align}

We now give useful a priori bounds for measures $\mathsf {q}$ and $\mathsf {R}$ .

Lemma 3.12. For $\mu = (\mathsf {G},\mathsf {q},\mathsf {R}) \in \mathcal {M}(\mathcal {X},\mathbb {X})$ with $\mathcal {J}_{\Lambda, \mathcal {X}}(\mu ) \lt +\infty$ , it holds that for $E \in \mathscr{B}(\mathcal {X})$ ,

(3.25) \begin{align} |\mathsf {q}|(E) \le \sqrt {Tr \mathsf {G} (E)}\, \lVert \Lambda _1 \rVert _{\mathrm {F}} \lVert G^\dagger q \Lambda _1^\dagger \rVert _{L^2_{\mathsf {G}}(E)}\,,\quad |\mathsf {R}|(E) \le \sqrt {Tr \mathsf {G} (E)}\, \lVert \Lambda _2 \rVert _{\mathrm {F}} \lVert G^\dagger R \Lambda _2^{-1} \rVert _{L^2_{\mathsf {G}}(E)}\,. \end{align}

Proof. Recall that there exist bounded measurable functions $\sigma _{q}$ and $\sigma _{R}$ with $ \lVert \sigma _{q} \rVert _{\mathrm {F}} = \lVert \sigma _{R} \rVert _{\mathrm {F}} = 1$ such that $\mathrm {d} \mathsf {q} = \sigma _{q}\, \mathrm {d} |\mathsf {q}|$ and $\mathrm {d} \mathsf {R} = \sigma _{R} \,\mathrm {d} |\mathsf {R}|$ . Taking $\mathsf {R} = 0$ and $(u,W) = (\chi _E \sigma _q, 0)$ in (3.23) for $E \in \mathscr {B}(\mathcal {X})$ , we obtain

\begin{align*} |\mathsf {q}|(E) = \int _E u \cdot \mathrm {d} \mathsf {q} \le 2 \sqrt {\mathcal {J}_{\Lambda, E}(\mathsf {G}, \mathsf {q}, 0)\mathcal {J}_{\Lambda, E}^{*}(\mathsf {G},u, 0)} \le \sqrt {Tr \mathsf {G} (E) \lVert \Lambda _1 \rVert _{\mathrm {F}}^2} \lVert G^\dagger q \Lambda _1^\dagger \rVert _{L^2_{\mathsf {G}}(E)} \,, \end{align*}

by (3.24) and the following estimate derived from (3.20) and (2.4):

\begin{align*} \mathcal {J}_{\Lambda, E}^*(\mathsf {G},u, W) \le \frac {1}{2} Tr \mathsf {G}(E) \lVert \Lambda _1 \rVert _{\mathrm {F}}^2\,. \end{align*}

Similarly, by taking $\mathsf {q} = 0$ and $(u,W) = (0, \chi _E \sigma _R)$ in (3.23), we obtain the estimate for $\mathsf {R}$ in (3.25).

With the help of the above lemma, the following proposition holds.

Proof. (i) First, note from [Reference Ambrosio, Gigli and Savaré1, Theorem 5.3.1] that $\mu$ can be disintegrated with respect to $\nu = \pi ^t_\# |\mu |$ as $\mu = \int _0^1 \delta _t \otimes \mu _t\, \mathrm {d} \nu$ , where $\mu _t \in \mathcal {M}(\Omega, \mathbb {X})$ for $\nu$ -a.e. $t \in [0,1]$ . Then, by Lemmas3.3 and 3.7, we have $\mathsf {G} \in \mathcal {M}(Q,\mathbb {S}^n_+)$ and $\nu \ll \pi ^t_\# | \mathsf {G}| \ll \mathrm {d} t$ on $[0,1]$ , which allows us to define $\widetilde {\mu }_t \,:\!=\, \mu _t \frac {\mathrm {d} \nu }{\mathrm {d} t}$ and disintegrate $\mu$ as $\mu = \int _0^1 \delta _t \otimes \widetilde {\mu }_t\, \mathrm {d} t$ .

(ii) Consider test functions $\Phi = a(t)\Psi (x)$ in (3.13) with $a(t) \in C_c^1((0,1),\mathbb {R})$ and $\Psi (x) \in C^1(\Omega, \mathbb {S}^n)$ . Then, by (3.26), $\int _{\Omega } \Psi \cdot \mathrm {d} \mathsf {G}_t$ is absolutely continuous in $t$ with the weak derivative:

(3.29) \begin{equation} \partial _t \langle \mathsf {G}_t, \Psi \rangle _{\Omega } = \langle \mathsf {q}_t, \mathsf {D}^* \Psi \rangle _{\Omega } + \langle \mathsf {R}_t, \Psi \rangle _{\Omega } \,. \end{equation}

Letting $\Psi = I$ in (3.29), we obtain $\partial _t Tr \mathsf {G}_t(\Omega ) = Tr \mathsf {R}^{\mathrm {sym}}_t(\Omega )$ a.e. by $\mathsf {D}^*(I) = 0$ , which implies that there exists a nonnegative function $m(t) \in C([0,1],\mathbb {R})$ such that $Tr \mathsf {G}_t (\Omega ) = m(t)$ a.e. on $[0,1]$ and

(3.30) \begin{equation} m(t) - m(s) = \int _s^t Tr \mathsf {R}^{\mathrm {sym}}_{\tau }(\Omega )\, \mathrm {d} \tau \,, \quad \forall 0\le s \le t \le 1\,. \end{equation}

By Lemma3.12, it follows from (3.30) that, from some $C \gt 0$ ,

(3.31) \begin{equation} |m(t) - m(s)| \le C |\mathsf {R}|(Q) \le C \sqrt {Tr \mathsf {G} (Q)} \lVert \Lambda _2 \rVert _{\mathrm {F}} \lVert G^\dagger R \Lambda _2^{-1} \rVert _{L^2_{\mathsf {G}}(Q)}\,. \end{equation}

We choose $t_0$ such that $m(t_0) = \max _{t \in [0,1]} m(t)$ . Then (3.31) implies

\begin{equation*} m(t_0) \le m(0) + C \sqrt {m(t_0)} \lVert \Lambda _2 \rVert _{\mathrm {F}} \lVert G^\dagger R \Lambda _2^{-1} \rVert _{L^2_{\mathsf {G}}(Q)}\,, \end{equation*}

which further gives, by an elementary calculation,

(3.32) \begin{align} \Big (m(t_0)^{1/2} - \frac {C}{2}\lVert G^\dagger R \Lambda _2^{-1} \rVert _{L_{\mathsf {G}}^2(Q)} \lVert \Lambda _2 \rVert _{\mathrm {F}}\Big )^2 \le m(0) + \frac {C^2}{4} \lVert G^\dagger R \Lambda _2^{-1} \rVert ^2_{L_{\mathsf {G}}^2(Q)} \lVert \Lambda _2 \rVert _{\mathrm {F}}^2\,. \end{align}

Then we have

(3.33) \begin{equation} m(t) \le C \big (m(0) + \lVert G^\dagger R \Lambda _2^{-1} \rVert ^2_{L_{\mathsf {G}}^2(Q)}\lVert \Lambda _2 \rVert _{\mathrm {F}}^2\big )\,. \end{equation}

With the above estimates, the existence of a weak^* continuous representative of $\mathsf {G}_t$ and the formula (3.27) can be proved similarly to [Reference Ambrosio, Gigli and Savaré1, Lemma 8.1.2]. We sketch the argument for completeness. By (3.25) and (3.33), as well as (3.29), there exists a subset $E \in [0,1]$ of Lebesgue measure zero such that $Tr \mathsf {G}_t (\Omega ) = m(t)$ on $[0,1]\backslash E$ , and there holds, for any $t,s \in [0,1]\backslash E$ with $s \lt t$ and $\Psi \in C^1(\Omega, \mathbb {S}^n)$ ,

(3.34) \begin{align} | \langle \mathsf {G}_t, \Psi \rangle _{\Omega } - \langle \mathsf {G}_s, \Psi \rangle _{\Omega }| & \le C \lVert \Psi \rVert _{1,\infty } \big (|\mathsf {q}|(Q_s^t) + |\mathsf {R}|(Q_s^t)\big ) \\ & \le C |t - s|^{1/2} \big (m(0) + \lVert G^\dagger q \Lambda _1^\dagger \rVert ^2_{L_{\mathsf {G}}^2(Q)} \lVert \Lambda _1 \rVert _{\mathrm {F}}^2 + \lVert G^\dagger R \Lambda _2^{-1} \rVert ^2_{L_{\mathsf {G}}^2(Q)}\lVert \Lambda _2 \rVert _{\mathrm {F}}^2\big ) \lVert \Psi \rVert _{1,\infty }\,.\nonumber \end{align}

The estimate (3.34) allows us to uniquely extend $\{\mathsf {G}_t\}_{t \in [0,1]\backslash E}$ to a weak^* continuous curve $\{\widetilde {\mathsf {G}}_t\}_{t \in [0,1]}$ in $C^1(\Omega, \mathbb {S}^n)^*$ . Then, by the density of $C^1(\Omega, \mathbb {S}^n)$ in $C(\Omega, \mathbb {S}^n)$ and the boundedness (3.33) of $\{Tr \widetilde {\mathsf {G}}_t (\Omega )\}_{t \in [0,1]}$ , the curve $\{\widetilde {\mathsf {G}}_t\}_{t \in [0,1]}$ is also weak^* continuous in $\mathcal {M}(\Omega, \mathbb {S}^n)$ . The formula (3.27) follows from taking test functions $\Phi _\varepsilon (x,t) = \eta _\varepsilon (t)\Phi (t,x)$ in (3.13), where $\Phi \in C^1(Q,\mathbb {S}^n)$ and $\eta _\varepsilon \in C_c^\infty ((t_0,t_1),\mathbb {R})$ with $0 \le \eta _\varepsilon \le 1$ , $\lim _{\varepsilon \to 0}\eta _\varepsilon (t) = \chi _{(t_0,t_1)}(t)$ pointwisely and $\lim _{\varepsilon \to 0}\eta^{\prime}_\varepsilon = \delta _{t_0} - \delta _{t_1}$ in the distributional sense. Recalling $Tr \mathsf {G}_t (\Omega ) = m(t)$ a.e., by the weak^* continuity of $\widetilde {\mathsf {G}}_t$ , we have $Tr \widetilde {\mathsf {G}}_t = m(t)$ . Then, the estimate (3.28) follows from (3.33).

3.5. Time and space scaling

By writing $\mathcal {J}_{\Lambda, Q}(\mu ) = \int _0^1\mathcal {J}_{\Lambda, \Omega }(\mu _t)\, \mathrm {d} t$ for $\mu \in \mathcal {CE}_\infty ([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ , the following Lemma is a simple consequence of the change of variable.

Remark 3.16. The condition (3.36) is nontrivial and necessary for the second statement. Indeed, there holds

\begin{align*} \mathsf{D}^{*} (\Phi \circ T) = \int _{\mathbb {R}^d} \widehat {\mathsf {D}^*}(\xi \cdot \nabla T(x))\big [\widehat {\Phi }(\xi )\big ]e^{\mathrm {i} \xi \cdot T(x)}\, \mathrm {d} \xi, \end{align*}

by Fourier transform, where $(\xi \cdot \nabla T(x))_j = \xi \cdot \partial _j T(x)$ . It follows that (3.36) is equivalent to a separation of variables: $\widehat {\mathsf {D}^*}(\xi \cdot \nabla T(x)) = \mathcal {T}_{\mathsf{D}^{*}}(x) \circ \widehat {\mathsf {D}^*}(\xi )$ . A sufficient condition for (3.36) is that $\widehat {\mathsf {D}^*}$ is homogeneous of degree $0$ , or homogeneous of degree $1$ with $T(x) = a x + b$ for $a \neq 0 \in \mathbb {R}$ and $b \in \mathbb {R}^d$ , which is enough for our purposes.

Remark 3.17. We connect the weight matrix $\Lambda _1$ and the space scaling. Let us consider $\mu \in \mathcal {CE}_\infty ([0,1]; \mathsf {G}_0, \mathsf {G}_1)$ and $\mathsf {D}^*$ be homogeneous of degree one for simplicity. Define $T(x) = a x\,:\, \Omega \to a \Omega$ and $\mathcal {T}_{\mathsf {D}} = a I$ . By Lemma 3.15, we have $\widetilde {\mu } \,:\!=\, \int _0^1 \delta _t \otimes T_{\#} (\mathsf {G}_{t}, a \mathsf {q}_{t}, \mathsf {R}_{t}) \, \mathrm {d} t \in \mathcal {CE}_\infty ([0,1];\, T_{\#}\mathsf {G}_0, T_{\#}\mathsf {G}_1)$ . Then, a direct computation gives

\begin{align*} \mathcal {J}_{\Lambda, [0,1] \times a \Omega }(\widetilde {\mu }) = \int _0^1 \mathcal {J}_{(a^{-1} \Lambda _1,\Lambda _2), a \Omega }(T_{\#}(\mathsf {G}_t,\mathsf {q}_t,\mathsf {R}_t)) \,\mathrm {d} t = \int _0^1 \mathcal {J}_{(a^{-1} \Lambda _1,\Lambda _2), \Omega }(\mu _t) \,\mathrm {d} t = \mathcal {J}_{(a^{-1} \Lambda _1,\Lambda _2), Q}(\mu ). \end{align*}

Using Lemma3.15 with $\mathsf {s}(t) = (b - a) t + a \,:\, [0,1] \to [a,b]$ , $b \gt a \gt 0$ , we see that for $\mu \in \mathcal {CE}_\infty ([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ , there exists $\widetilde {\mu } \in \mathcal {CE}_\infty ([a,b];\mathsf {G}_0,\mathsf {G}_1)$ such that

\begin{align*} \int _0^1 \mathcal {J}_{\Lambda, \Omega }(\mu _t)\, \mathrm {d} t = (b - a) \int _a^b \mathcal {J}_{\Lambda, \Omega }(\widetilde {\mu }_t)\, \mathrm {d} t\,, \end{align*}

and vice versa, which gives the equivalent characterisation of $\mathrm {WB}_{\Lambda }$ :

(𝒫^′) \begin{align} \mathrm {WB}^2_\Lambda (\mathsf {G}_0,\mathsf {G}_1) = \inf _{\mathcal {CE}_\infty ([a,b];\mathsf {G}_0,\mathsf {G}_1)} (b - a) \int _a^b \mathcal {J}_{\Lambda, \Omega }(\mu _t) \,\mathrm {d} t\,, \quad \mathsf {G}_0,\mathsf {G}_1 \in \mathcal {M}(\Omega, \mathbb {S}_+^n)\,. \end{align}

3.6. Compactness

We end the discussion of basic properties of $\mathcal {CE}_\infty ([0,1];\, \mathsf {G}_0, \mathsf {G}_1)$ with a compactness result.

Proposition 3.18. Let $\mu ^n = (\mathsf {G}^n,\mathsf {q}^n,\mathsf {R}^n) \in \mathcal {CE}_\infty ([0,1];\, \mathsf {G}^n_0, \mathsf {G}^n_1)$ , $n \ge 1$ , be a sequence of measures satisfying

(3.37) \begin{equation} m\,:\!=\, \sup _{n \in \mathbb {N}} Tr (\mathsf {G}_0^n) \lt +\infty \,, \quad M\,:\!=\, \sup _{n \in \mathbb {N}} \mathcal {J}_{\Lambda, Q}(\mu ^n) \lt +\infty \,. \end{equation}

Then there exists a subsequence, still denoted by $\mu ^n$ , and a measure $\mu = (\mathsf {G},\mathsf {q},\mathsf {R}) \in \mathcal {CE}_\infty ([0,1]; \mathsf {G}_0, \mathsf {G}_1)$ such that for every $t\in [0,1]$ , $\mathsf {G}^n_t$ weak^* converges to $\mathsf {G}_t$ in $\mathcal {M}(\Omega, \mathbb {S}^{n})$ , and $ (\mathsf{q}^{\mathsf{n}}, \mathsf{R}^{\mathsf{n}})$ weak^* converges to $\mathsf {(q,R)}$ in $\mathcal {M}(Q,\mathbb {R}^{n \times k} \times \mathbb {M}^n)$ . Moreover, it holds that, for $0\le a \lt b \le 1$ ,

(3.38) \begin{align} \mathcal {J}_{\Lambda, Q_a^b}(\mu ) \le \liminf _{n \to \infty } \mathcal {J}_{\Lambda, Q_a^b}(\mu ^n)\,. \end{align}

Proof. By (3.37), up to a subsequence, we can let $\mathsf {G}^n_0$ weak^* converge to some $\mathsf {G}_0 \in \mathcal {M}(\Omega, \mathbb {S}^n_+)$ . It is also clear from a priori estimates (3.25) and (3.28), as well as the assumption (3.37), that $\{\mu ^n\}_{n \in \mathbb {N}}$ is bounded in $\mathcal {M}(Q,\mathbb {X})$ . Hence, there exists a subsequence of $\{\mu ^n\}_{n \in \mathbb {N}}$ , still indexed by $n$ , weak^* converging to some $\mu \in \mathcal {M}(Q,\mathbb {X})$ . We next prove that the restriction of $\mu ^n$ on $Q_a^b$ , i.e., $\mu ^n|_{Q_a^b}$ , weak^* converges to $\mu |_{Q_a^b}$ in $\mathcal {M}(Q_a^b,\mathbb {X})$ for any $ 0 \le a \le b \le 1$ . For this, again by (3.25) and (3.28), we have, for some $C \gt 0$ ,

(3.39) \begin{equation} |\mu ^n|([t_0,t_1] \times \Omega ) \le C|t_1 - t_0|^{1/2}\,, \quad \forall 0 \le t_0 \le t_1 \le 1\,, \end{equation}

which also holds for $\mu$ . Let $\eta (t)$ be a smooth function, compactly supported in $[a,b]$ , with $|\eta (t)| \le 1$ and $\eta = 1$ on $[a+\varepsilon, b - \varepsilon ]$ for some small $\varepsilon$ . Then, for any $\Xi \in C(Q_a^b,\mathbb {X})$ , we define $\widetilde {\Xi }(t,x) = \eta (t) \Xi (t,x) \in C(Q,\mathbb {X})$ . The following estimate readily follows from the properties of $\eta$ and the estimate (3.39):

\begin{equation*} \big |\langle \mu ^n, \Xi \rangle _{Q_a^b} - \langle \mu, \Xi \rangle _{Q_a^b}\big | \le \big | \big \langle \mu ^n, \widetilde {\Xi }\big \rangle _{Q} - \big \langle \mu, \widetilde {\Xi }\big \rangle _{Q} \big | + C \varepsilon ^{1/2}\,. \end{equation*}

Since $\mu ^n$ weak^* converges to $\mu$ in $\mathcal {M}(Q,\mathbb {X})$ and $\varepsilon$ is arbitrary, we have $\big |\langle \mu ^n, \Xi \rangle _{Q_a^b} - \langle \mu, \Xi \rangle _{Q_a^b}\big | \to 0$ as $n \to \infty$ for $\Xi \in C(Q_a^b,\mathbb {X})$ . Then, (3.38) follows from the lower semicontinuity of $\mathcal {J}_{\Lambda, Q_a^b}(\mu )$ . We now show the weak^* convergence of $\mathsf {G}^n_t$ for every $t\in [0,1]$ . We note, by taking $\Phi (s,x) = \chi _{[0,t]}(s) \Psi (x)$ in (3.27) with $\Psi (x)\in C^1(\Omega, \mathbb {S}^n)$ ,

\begin{equation*} \int _0^t \Big ( \int _\Omega \mathsf {D}^* \Psi \cdot \mathrm {d} \mathsf {q}_s^n + \int _\Omega \Psi \cdot \mathrm {d} \mathsf {R}_s^n \Big ) \mathrm {d} s = \int _{\Omega } \Psi \cdot \mathrm {d} \mathsf {G}^n_{t} - \int _{\Omega } \Psi \cdot \mathrm {d} \mathsf {G}^n_{0}\,, \quad \forall \Psi \in C^1(\Omega, \mathbb {S}^n)\,. \end{equation*}

Then, using the weak^* convergences of $\mathsf {G}^n_0$ in $\mathcal {M}(\Omega, \mathbb {S}^n)$ and $(\mathsf {q}^n,\mathsf {R}^n)|_{Q_0^t}$ in $\mathcal {M}(Q_0^t,\mathbb {R}^{n \times k}\times \mathbb {M}^n)$ , we get the convergence of $ \langle \mathsf {G}^n_{t}, \Psi \rangle _{\Omega }$ as $n \to \infty$ . The proof is completed by the density of $C^1(\Omega, \mathbb {S}^n)$ in $C(\Omega, \mathbb {S}^n)$ and the uniform boundedness of $Tr \mathsf {G}^n_t (\Omega )$ with respect to $n$ from (3.28).

4.1. Existence of minimiser and optimality condition

For our purpose, let us first define the Lagrangian of (𝒫) with the multiplier $\Phi \in C^1(Q, \mathbb {S}^n)$ :

\begin{align*} \mathcal {L}(\mu, \Phi ) \,:\!=\, \mathcal {J}_{\Lambda, Q}(\mu ) - \langle \mu, (\partial _t \Phi, \mathsf {D}^* \Phi, \Phi ) \rangle _Q + \langle \mathsf {G}_1, \Phi _1 \rangle _\Omega - \langle \mathsf {G}_0, \Phi _0 \rangle _\Omega \,, \end{align*}

which allows us to write

\begin{equation*} \mathrm {WB}^2_\Lambda (\mathsf {G}_0,\mathsf {G}_1) = \inf _{\mu \in \mathcal {M}(Q,\mathbb {X})} \sup _{\Phi \in C^1(Q,\mathbb {S}^n)} \mathcal {L}(\mu, \Phi )\,. \end{equation*}

By changing the order of $\sup$ and $\inf$ , a formal calculation via integration by parts gives the dual problem:

(4.1) \begin{align} \mathrm {WB}_{\Lambda }^2(\mathsf {G}_0,\mathsf {G}_1) & \ge \sup _\Phi \inf _\mu \mathcal {L}(\mu, \Phi ) \notag \\ & = \sup _{\Phi } \Big \{ \langle \mathsf {G}_1, \Phi _1 \rangle _\Omega - \langle \mathsf {G}_0, \Phi _0 \rangle _\Omega \,; \ \partial _t \Phi + \frac {1}{2} (\mathsf {D}^* \Phi ) \Lambda _1^2 (\mathsf {D}^* \Phi )^{\mathrm {T}} + \frac {1}{2} \Phi \Lambda _2^2 \Phi \preceq 0 \Big \}\,. \end{align}

We next use the Fenchel-Rockafellar theorem (Lemma2.5) to show that the duality gap is zero, which will also give the existence of the minimiser to (𝒫) and the optimality conditions. For this, we define

(4.2) \begin{equation} C(Q,\mathcal {O}_\Lambda ) \,:\!=\, \{\varphi \in C(Q,\mathbb {X})\,;\ \varphi (x) \in \mathcal {O}_\Lambda \,, \ \forall x \in Q\}\,, \end{equation}

with $\mathcal {O}_\Lambda$ given in (3.1), which is a closed convex subset of $C(Q,\mathbb {X})$ . We then define lower semicontinuous convex functions: $f(\Phi ) = \langle \mathsf {G}_1, \Phi _1 \rangle _\Omega - \langle \mathsf {G}_0, \Phi _0 \rangle _\Omega$ for $\Phi \in C^1(Q,\mathbb {S}^n)$ and $g(\Xi ) = \iota _{C(Q,\mathcal {O}_\Lambda )}(\Xi )$ for $\Xi \in C(Q,\mathbb {X})$ . We also introduce the bounded linear operator: $L\,:\, \Phi \in C^1(Q, \mathbb {S}^n) \to (\partial _t \Phi, \mathsf {D}^* \Phi, \Phi ) \in C(Q,\mathbb {X})$ with the dual operator $L^*$ . These notions help us to write (4.1) as $ \sup \{f(\Phi ) - g(L \Phi )\,;\ \Phi \in C^1(Q,\mathbb {S}^n)\}\,.$

We now verify the condition in Lemma2.5. We consider $\Phi = -\varepsilon t I + \frac {\varepsilon }{2} I \in C^1(Q,\mathbb {S}^n)$ . It is clear that $f(\Phi )$ is finite and $L \Phi = ({-} \varepsilon I, 0, -\varepsilon t I + \frac {\varepsilon }{2} I)$ by $\mathsf {D}^*(I) = 0$ . By a simple calculation, we have

\begin{align*} \partial _t \Phi + \frac {1}{2} (\mathsf {D}^*\Phi ) \Lambda _1^2 (\mathsf {D}^* \Phi )^{\mathrm {T}} + \frac {1}{2} \Phi \Lambda _2^2 \Phi &= - \varepsilon I+ \frac {1}{2}\varepsilon ^2 \Big({-} t + \frac {1}{2}\Big)^2 \Lambda _2^2 \preceq - \varepsilon I+ \frac {1}{8} \varepsilon ^2 \Lambda _2^2\,, \end{align*}

which implies that for small enough $\varepsilon$ and any $(t,x)\in Q$ , $(L \Phi )(t,x)$ is in the interior of $\mathcal {O}_\Lambda$ and hence $g$ is continuous at $L \Phi$ . Then Lemma2.5 readily gives

(4.3) \begin{equation} \min _{\mu \in \mathcal {M}(Q,\mathbb {X})}f^*(L^*\mu ) + g^*(\mu ) = \sup _{\Phi \in C^1(Q,\mathbb {S}^n)} f(\Phi ) - g(L \Phi )\,, \end{equation}

where $f^*(L^*\mu ) = \sup \{\langle \mu, L\Phi \rangle _Q - f(\Phi )\,;\ \Phi \in C^1(Q,\mathbb {S}^n)\}$ can be easily computed as $\iota _{\mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)}$ by linearity of $f$ , while $g^*(\mu )$ is nothing else than $\mathcal {J}_{\Lambda, Q}(\mu )$ by the following lemma, which is a direct application of general results [Reference Bouchitté and Valadier13, Reference Rockafellar83]. We sketch the proof in Appendix A for completeness.

Lemma 4.1. Let $\mathcal {X}$ be a compact separable metric space and $C(\mathcal {X},\mathcal {O}_\Lambda )$ be defined in (4.2). Then, we have

(4.4) \begin{equation} \iota ^*_{C(\mathcal {X},\mathcal {O}_\Lambda )} = \sup _{\Xi \in L_{|\mu |}^\infty (\mathcal {X},\mathcal {O}_\Lambda )}\langle \mu, \Xi \rangle _{\mathcal {X}} = \mathcal {J}_{\Lambda, \mathcal {X}} (\mu )\,,\quad \text {for}\ \mu \in \mathcal {M}(\mathcal {X},\mathbb {X})\,, \end{equation}

which is proper convex and lower semicontinuous with respect to the weak^* topology of $\mathcal {M}(\mathcal {X},\mathbb {X})$ . Moreover, the subgradient $\partial \mathcal {J}_{\Lambda, \mathcal {X}}(\mu )$ in $C(\mathcal {X},\mathbb {X})$ is given as follows:

(4.5) \begin{equation} \partial \mathcal {J}_{\Lambda, \mathcal {X}}(\mu )|_{C(\mathcal {X},\mathbb {X})} = \left \{\Xi \in C(\mathcal {X}, \mathcal {O}_\Lambda )\,; \ \Xi (x) \in \partial J_\Lambda (\mu _\lambda )(x)\,, \ \lambda \text{-a.e.}\right \}\,, \end{equation}

which is independent of the choice of the reference measure $\lambda$ such that $|\mu | \ll \lambda$ .

By the above arguments, we have shown the following result.

Theorem 4.2. The optimisation problem ( 𝒫 ) always admits a minimiser $\mu \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ and a dual formulation with zero duality gap:

(4.6) \begin{equation} \mathrm {WB}^2_{\Lambda }(\mathsf {G}_0,\mathsf {G}_1) = \sup _{\Phi \in C^1(Q,\mathbb {S}^n)} \left \{ \langle \mathsf {G}_1, \Phi _1 \rangle _{\Omega } - \langle \mathsf {G}_0, \Phi _0 \rangle _\Omega - \iota _{C(Q,\mathcal {O}_\Lambda )}(\partial _t \Phi, \mathsf {D}^* \Phi, \Phi ) \right \}\,, \end{equation}

where the $\sup$ is attained at $\Phi \in C^1(Q,\mathbb {S}^n)$ if and only if there exists $\mu = \mathsf {(G,q,R)} \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ such that

(4.7) \begin{align} q_\lambda = G_\lambda (\mathsf {D}^* \Phi ) \Lambda _1^2 \,, \quad R_\lambda = G_\lambda \Phi \Lambda _2^2\,, \end{align}

and

(4.8) \begin{equation} G_\lambda \cdot \Big (\partial _t \Phi + \frac {1}{2} (\mathsf {D}^* \Phi ) \Lambda _1^2 (\mathsf {D}^* \Phi )^{\mathrm {T}} + \frac {1}{2} \Phi \Lambda _2^2 \Phi \Big ) = 0 \,, \end{equation}

for $\lambda$ -a.e. $(t,x) \in Q$ . In this case, $\mu$ is also the minimiser to the problem ( $\mathcal {P}$ ).

As a consequence of Lemma4.1 and the dual formulation (4.6), we have the sublinearity and the weak^* lower semicontinuity of $\mathrm {WB}^2_{\Lambda }(\cdot, \cdot )$ .

Corollary 4.3. $\mathrm {WB}^2_{\Lambda }(\cdot, \cdot )$ is sublinear: for $\alpha \gt 0$ , $\mathsf {G}_0,\mathsf {G}_1,\widetilde {\mathsf {G}}_0,\widetilde {\mathsf {G}}_1 \in \mathcal {M}(\Omega, \mathbb {S}^n_+)$ , there holds

(4.9) \begin{align} \mathrm {WB}^2_{\Lambda }\big (\alpha \mathsf {G}_0, \alpha \mathsf {G}_1\big ) = \alpha \mathrm {WB}^2_{\Lambda }\big (\mathsf {G}_0, \mathsf {G}_1\big )\,,\quad \mathrm {WB}^2_{\Lambda }\big (\mathsf {G}_0 +\widetilde {\mathsf {G}}_0, \mathsf {G}_1 + \widetilde {\mathsf {G}}_1\big ) \le \mathrm {WB}^2_{\Lambda }\big (\mathsf {G}_0, \mathsf {G}_1\big ) + \mathrm {WB}^2_{\Lambda }\big (\widetilde {\mathsf {G}}_0, \widetilde {\mathsf {G}}_1\big )\,. \end{align}

Moreover, $\mathrm {WB}_{\Lambda }$ is lower semicontinuous with respect to the weak^* topology, that is, for any sequences $\{\mathsf {G}^n_0\}_{n \in \mathbb {N}}$ and $\{\mathsf {G}^n_1\}_{n \in \mathbb {N}}$ in $\mathcal {M}(\Omega, \mathbb {S}_+^n)$ that weak^* converge to measures $\mathsf {G}_0, \mathsf {G}_1 \in \mathcal {M}(\Omega, \mathbb {S}_+^n)$ , respectively, there holds

(4.10) \begin{align} \mathrm {WB}_{\Lambda } (\mathsf {G}_0,\mathsf {G}_1) \le \liminf _{n \to 0} \mathrm {WB}_{\Lambda } (\mathsf {G}^n_0,\mathsf {G}^n_1)\,. \end{align}

Proof. Noting that $\mathcal {J}_{\Lambda, Q}(\mu )$ is positively homogeneous and convex, and hence sublinear, the sublinearity of $\mathrm {WB}^2_{\Lambda }(\cdot, \cdot )$ follows from definition ( $\mathcal {P}$ ) and the linearity of the continuity equation. For the weak^* lower semicontinuity, by (4.6), for any $\Phi \in C^1(Q,\mathbb {S}^n)$ with $\iota _{C(Q,\mathcal {O}_\Lambda )}(\partial _t \Phi, \mathsf {D}^* \Phi, \Phi ) = 0$ , there holds

(4.11) \begin{align} \liminf _{n \to \infty } \mathrm {WB}^2_{\Lambda }(\mathsf {G}^n_0,\mathsf {G}^n_1) \ge \liminf _{n \to \infty } \langle \mathsf {G}^n_1, \Phi _1 \rangle _{\Omega } - \langle \mathsf {G}^n_0, \Phi _0 \rangle _\Omega = \langle \mathsf {G}_1, \Phi _1 \rangle _{\Omega } - \langle \mathsf {G}_0, \Phi _0 \rangle _\Omega \,, \end{align}

by the weak^* convergence of $\mathsf {G}_0^n$ and $\mathsf {G}_1^n$ . Then (4.10) follows by taking the $\sup$ of (4.11) over admissible $\Phi$ .

In addition, we have the following explicit characterisation of the minimiser (i.e., geodesic; see Corollary5.7) to ( $\mathcal {P}$ ) for inflating measures from optimality conditions (4.7) and (4.8), which extends [Reference Brenier and Vorotnikov16, Theorem 5] with a much simpler argument. For $\mathsf {G} \in \mathcal {M}(\Omega, \mathbb {S}^n_+)$ and $A \in \mathbb {S}_+^n$ , we denote by $\mathsf {G}^A$ the inflating measure $A \mathsf {G} A \in \mathcal {M}(\Omega, \mathbb {S}^n_+)$ .

Proposition 4.4. For $\mathsf {G} \in \mathcal {M}(\Omega, \mathbb {S}^n_+)$ and matrices $A_0, A_1 \in \mathbb {S}_+^n$ , we have

(4.12) \begin{equation} \mathrm {WB}_\Lambda ^2 \big (\mathsf {G}^{A_0},\mathsf {G}^{A_1}\big ) = 2 Tr \big (\Lambda _2^{-1}(A_1 - A_0)\mathsf {G}(\Omega )(A_1 - A_0)\Lambda _2^{-1} \big )\,, \end{equation}

with the minimiser $(\mathsf {G}_*,\mathsf {q}_*,\mathsf {R}_*) \,:\!=\, (\mathsf {G}^{A_t}, 0, 2 A_t \mathsf {G} (A_1 - A_0)) \in \mathcal {M}(Q, \mathbb {X})$ , where $A_t \,:\!=\, tA_1 + (1- t)A_0$ for $t \in [0,1]$ .

Proof. Let us first assume that $A_0$ and $A_1$ are invertible. By a direct calculation, we have

\begin{equation*} \partial _t \mathsf {G}^{\mathsf {A}_t} = (A_1 - A_0) \mathsf {G} A_t + A_t \mathsf {G} (A_1 - A_0) \,. \end{equation*}

We define $\Phi = 2 A_t^{-1} (A_1- A_0)\Lambda _2^{-2}$ and find $\mathsf {R}_* = \mathsf {G}^{A_t} \Phi \Lambda _2^2$ . It is also easy to see that $(\mathsf {G}_*,\mathsf {q}_*,\mathsf {R}_*)$ defined above is in the set $\mathcal {CE}\big ([0,1]; \mathsf {G}^{A_0},\mathsf {G}^{A_1}\big )$ . Moreover, recalling $ ((A + \varepsilon H)^{-1} - A^{-1})/\varepsilon \to - A^{-1}H A^{-1}$ as $\varepsilon \to 0$ for invertible $A$ and $H \in \mathbb {M}^n$ [Reference Bhatia9], we have

\begin{equation*} \partial _t \Phi = - 2 A_t^{-1} (A_1 - A_0)A_t^{-1}(A_1- A_0) \Lambda _2^{-2} = - \Phi \Lambda _2^2 \Phi /2\,. \end{equation*}

By the above computations, we have verified the optimality conditions (4.7) and (4.8), which means that the measure $(\mathsf {G}_*,\mathsf {q}_*,\mathsf {R}_*)$ is the desired minimiser. Then, we can further compute

\begin{equation*} \mathrm {WB}_\Lambda ^2\big (\mathsf {G}^{A_0}, \mathsf {G}^{A_1}\big ) = \frac {1}{2} \int _0^1 \int _\Omega (\Phi \Lambda _2) \cdot \mathrm {d} \mathsf {G}^{A_t} (\Phi \Lambda _2)\, \mathrm {d} t = 2 ((A_1 - A_0)\Lambda _2^{-1})\cdot \mathsf {G}(\Omega )(A_1 - A_0)\Lambda _2^{-1}\,. \end{equation*}

For general $A_0,A_1 \in \mathbb {S}_+^n$ , we first see that $\mu _* \,:\!=\, (\mathsf {G}^{A_t}, 0, 2 A_t \mathsf {G} (A_1 - A_0))$ as above still satisfies the continuity equation and its associated action functional $\mathcal {J}_{\Lambda, Q}(\mu _*)$ gives the right-hand side of (4.12) by $\mathrm {Ran}(A_1 - A_0) \subset \mathrm {Ran}(A_t)$ , which also means $ \mathrm {WB}_\Lambda ^2(\mathsf {G}^{A_0}, \mathsf {G}^{A_1}) \le \mathcal {J}_{\Lambda, Q}(\mu _*)$ . To finish the proof, it suffices to show that the equality holds. For this, we consider $A_i^\varepsilon = A_i + \varepsilon I \in \mathbb {S}_{++}^n$ for $i = 0,1$ . Then, by triangle inequality of $\mathrm {WB}_\Lambda$ (see Proposition5.2 below) and Lemma3.9, we have $\mathrm {WB}_\Lambda (\mathsf {G}^{A^\varepsilon _0}, \mathsf {G}^{A^\varepsilon _1}) \to \mathrm {WB}_\Lambda (\mathsf {G}^{A_0}, \mathsf {G}^{A_1})$ as $\varepsilon \to 0$ . The proof is completed by

\begin{align*} \mathrm {WB}^2_\Lambda \big (\mathsf {G}^{A^\varepsilon _0}, \mathsf {G}^{A^\varepsilon _1}\big ) = & 2 Tr \big (\Lambda _2^{-1}(A^\varepsilon _1 - A^\varepsilon _0)\mathsf {G}(\Omega )(A^\varepsilon _1 - A^\varepsilon _0)\Lambda _2^{-1} \big )\\ & \to 2 Tr \big (\Lambda _2^{-1}(A_1 - A_0)\mathsf {G}(\Omega )(A_1 - A_0)\Lambda _2^{-1} \big ) = \mathcal {J}_{\Lambda, Q}(\mu _*) \,, \quad \varepsilon \to 0\,. \end{align*}

4.2. Primal-dual formulations

We proceed to study in more depth the optimality conditions by viewing $\mathsf {G}$ as the main variable and $\mathsf {(q,R)}$ as the control variable, which will be useful in Section 5. We first observe

(4.13) \begin{align} \mathrm {WB}^2_{\Lambda }(\mathsf {G}_0,\mathsf {G}_1) & = \inf _{\mathsf {G}} \inf _{\mathsf {q},\mathsf {R}} \left \{ \mathcal {J}_{\Lambda, Q}(\mu ) \,;\ \mu = (\mathsf {G}, \mathsf {q}, \mathsf {R}) \in \mathcal {CE}_\infty ([0,1];\,\mathsf {G}_0,\mathsf {G}_1) \right \}\,, \end{align}

by taking the inf in ( $\mathcal {P}$ ) over $\mathsf {G}$ and $(\mathsf {q},\mathsf {R})$ separately. Recall the formulation (3.24) of $\mathcal {J}_{\Lambda, Q}(\mu )$ , which motivates us to introduce a weighted semi-inner product:

(4.14) \begin{align} \big \langle (u,W), (u^{\prime},W^{\prime}) \big \rangle _{L^2_{\mathsf {G},\Lambda }(Q)} \,:\!=\, \big \langle u \Lambda _1^{\dagger }, u^{\prime} \Lambda _1^{\dagger } \big \rangle _{L^2_{\mathsf {G}}(Q)} + \big \langle W \Lambda _2^{-1}, W^{\prime} \Lambda _2^{-1} \big \rangle _{L^2_{\mathsf {G}}(Q)} \,, \end{align}

and the associated seminorm $\lVert \cdot \rVert _{L^2_{\mathsf {G},\Lambda }(Q)}$ on the space of measurable functions valued in $\mathbb {R}^{n \times k} \times \mathbb {M}^n$ . The corresponding Hilbert space, denoted by $L^2_{\mathsf {G},\Lambda }(Q,\mathbb {R}^{n \times k} \times \mathbb {M}^n)$ , is defined as the quotient space by the subspace $\mathrm {Ker}\big (\lVert \cdot \rVert _{L^2_{\mathsf {G},\Lambda }(Q)}\big )$ . Hence, we can rewrite (3.24) as $\mathcal {J}_{\Lambda, Q}(\mu ) = \lVert (G^\dagger q,G^\dagger R) \rVert ^2_{L^2_{\mathsf {G},\Lambda }(Q)}/2$ . Moreover, we define the set

(4.15) \begin{equation} \mathcal {AC}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)\,:\!=\, \{\mathsf {G} \in \mathcal {M}(Q,\mathbb {S}^n)\,; \ \exists \mathsf {(q,R)} \in \mathcal {M}(Q,\mathbb {R}^{n \times k} \times \mathbb {M}^n) \ \text {s.t.}\ \mathsf {(G,q,R)} \in \mathcal {CE}_\infty ([0,1];\,\mathsf {G}_0,\mathsf {G}_1)\}\,, \end{equation}

and the associated energy functional: for $\mathsf {G} \in \mathcal {AC}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ ,

(4.16) \begin{align} \mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}(\mathsf {G}) \,:\!=\, \inf _{\mathsf {(q,R)}} \Big \{\frac {1}{2} \lVert (G^\dagger q, G^\dagger R ) \rVert ^2_{L^2_{\mathsf {G},\Lambda }(Q)}\,;\ \mathsf {(G,q,R)} \in \mathcal {CE}_\infty ([0,1];\,\mathsf {G}_0,\mathsf {G}_1) \Big \}\,. \end{align}

We will see in Remark5.6 that $\mathcal {AC}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ is closely related to the set of absolutely continuous curves in the metric space $(\mathcal {M}(\Omega, \mathbb {S}^n_+), \mathrm {WB}_{\Lambda })$ . With the help of these notions, (4.13) can be reformulated in a compact form:

(4.17) \begin{equation} \mathrm {WB}^2_{\Lambda }(\mathsf {G}_0,\mathsf {G}_1) = \inf _{\mathsf {G}\in \mathcal {AC}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)} \mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}(\mathsf {G})\,. \end{equation}

Similarly to (3.24), by Lemma3.3, we also note that for $\mathsf {(G,q,R)} \in \mathcal {CE}_\infty ([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ , the weak formulation (3.13) can be written as

(4.18) \begin{equation} \big \langle \big (\mathsf {D}^* \Phi \Lambda ^2_1, \Phi \Lambda _2^2 \big ), \big (G^\dagger q, G^\dagger R\big ) \big \rangle _{L^2_{\mathsf {G}, \Lambda }(Q)} = l_{\mathsf {G}}(\Phi )\,, \quad \forall \Phi \in C^1(Q,\mathbb {S}^n)\,, \end{equation}

where $l_{\mathsf {G}}(\cdot )$ for $\mathsf {G} \in \mathcal {AC}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ is a linear functional on $C^1(Q,\mathbb {S}^n)$ defined by

(4.19) \begin{equation} l_{\mathsf {G}}(\Phi ) = \langle \mathsf {G}_1, \Phi _1 \rangle _{\Omega } - \langle \mathsf {G}_0, \Phi _0 \rangle _{\Omega } - \langle \mathsf {G}, \partial _t\Phi \rangle _Q \,. \end{equation}

Define an injective map $\Pi \,:\, \Phi \to (\mathsf {D}^* \Phi \Lambda _1^2, \Phi \Lambda _2^2)$ for $\Phi \in C^1(Q,\mathbb {S}^n)$ and denote $\widetilde {l}_{\mathsf {G}}\,:\!=\, l_{\mathsf {G}} \circ \Pi ^{-1}$ on the image of $\Pi$ . In view of (4.18), the functional $\widetilde {l}_{\mathsf {G}}$ can be uniquely extended to the space

(4.20) \begin{equation} H_{\mathsf {G},\Lambda }(\mathsf {D}^*)\,:\!=\, \overline {\left \{ \Pi (\Phi ) \,;\ \Phi \in C^1(Q,\mathbb {S}^n)\right \}}^{\lVert \cdot \rVert _{L^2_{\mathsf {G},\Lambda }(Q)}}\,, \end{equation}

with the norm estimate

(4.21) \begin{equation} \lVert \widetilde {l}_{\mathsf {G}} \rVert _{H^*_{\mathsf {G},\Lambda }(\mathsf {D}^*)} \le \lVert (G^\dagger q, G^\dagger R) \rVert _{L^2_{\mathsf {G},\Lambda }(Q)}\,. \end{equation}

We emphasise that such an extension is independent of the choice of $\mathsf {(q,R)}$ that satisfies $\mathsf {(G,q,R)} \in \mathcal {CE}_\infty ([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ .

Next, we show that (4.16) admits a unique minimiser $\mathsf {(q,R)}$ that satisfies the equality in (4.21). Note that $(u,W)$ and $(u \mathbb {P}_{\Lambda _1}, W \mathbb {P}_{\Lambda _2})$ are equivalent in $L^2_{\mathsf {G},\Lambda }(Q,\mathbb {R}^{n \times k} \times \mathbb {M}^n)$ , where $\mathbb {P}_{\Lambda _i}$ is the orthogonal projection to $\mathrm {Ran}(\Lambda _i)$ . Hence, for any $(u,W) \in L^2_{\mathsf {G},\Lambda }(Q,\mathbb {R}^{n \times k} \times \mathbb {M}^n)$ , we can assume $\mathrm {Ran}(u^{\mathrm {T}}) \subset \mathrm {Ran} (\Lambda _1)$ and $\mathrm {Ran}(W^{\mathrm {T}}) \subset \mathrm {Ran} (\Lambda _2)$ . Then, it holds that any $L^2_{\mathsf {G},\Lambda }$ -field $(u,W)$ satisfying $ \langle (\mathsf {D}^* \Phi \Lambda _1^2, \Phi \Lambda _2^2), (u,W)\rangle _{L^2_{\mathsf {G},\Lambda }(Q)} = l_{\mathsf {G}}(\Phi )$ , $\forall \Phi \in C^1(Q,\mathbb {S}^n)$ , induces a measure $\mathsf {(q,R)} \,:\!=\, (\mathsf {G} u,\mathsf {G}W)$ such that $\mathsf {(G,q,R)} \in \mathcal {CE}_\infty ([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ . This observation implies that $\mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}(\mathsf {G})$ is actually a uniquely solvable minimum norm problem with an affine constraint:

(4.22) \begin{align} \mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}(\mathsf {G}) &= \inf \Big \{\frac {1}{2} \lVert (u,W) \rVert ^2_{L^2_{\mathsf {G},\Lambda }(Q)}\,;\ (u,W) \in L^2_{\mathsf {G},\Lambda }(Q,\mathbb {R}^{n \times k} \times \mathbb {M}^n)\ \text {such that} \notag \\ \big \langle (\mathsf {D}^* \Phi \Lambda _1^2, \Phi \Lambda _2^2), (u,W) \big \rangle _{L^2_{\mathsf {G},\Lambda }(Q)} &= l_{\mathsf {G}}(\Phi )\,, \ \forall \Phi \in C^1(Q,\mathbb {S}^n) \Big \}\,. \end{align}

The unique minimiser $(u_*,W_*)$ to (4.22) is given by the orthogonal projection of $0$ on the constraint set, equivalently, the Riesz representation of the functional $\widetilde {l}_{\mathsf {G}}$ on the space $H_{\mathsf {G},\Lambda }(\mathsf {D}^*)$ . It then follows that $(\mathsf {q}_*,\mathsf {R}_*) \,:\!=\, (\mathsf {G} u_*,\mathsf {G}W_*)$ is the desired minimiser to (4.16) and there holds

(4.23) \begin{align} \lVert \widetilde {l}_{\mathsf {G}} \rVert _{H^*_{\mathsf {G},\Lambda }(\mathsf {D}^*)} & = \lVert (u_*,W_*) \rVert _{L^2_{\mathsf {G},\Lambda }(Q)} = \lVert (G^\dagger q_*, G^\dagger R_*) \rVert _{L^2_{\mathsf {G},\Lambda }(Q)}\,. \end{align}

We summarise the above facts in the following useful result.

Theorem 4.5. $\mathrm {WB}^2_{\Lambda }(\mathsf {G}_0,\mathsf {G}_1)$ has the following representation:

\begin{equation*} \mathrm {WB}^2_{\Lambda }(\mathsf {G}_0,\mathsf {G}_1) = \inf _{\mathsf {G}\in \mathcal {AC}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)} \mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}(\mathsf {G})\quad \text {with}\quad \mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}(\mathsf {G}) = \frac {1}{2}\lVert (u_*, W_*) \rVert ^2_{L^2_{\mathsf {G},\Lambda }(Q)}\,, \end{equation*}

where $(u_*, W_*)$ is the Riesz representation of $\widetilde {l}_{\mathsf {G}}$ in $H_{\mathsf {G},\Lambda }(\mathsf {D}^*)$ that uniquely solves the minimum norm problem (4.22).

Moreover, $\mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}(\mathsf {G})$ admits the following dual formulation:

(4.24) \begin{equation} \mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}(\mathsf {G}) = \sup \Big \{l_{\mathsf {G}}(\Phi ) - \frac {1}{2}\lVert (\mathsf {D}^* \Phi \Lambda _1^2,\Phi \Lambda _2^2) \rVert ^2_{L^2_{\mathsf {G},\Lambda }(Q)}\,; \ \Phi \in C^1(Q,\mathbb {S}^n) \Big \}\,. \end{equation}

Proof. It suffices to derive the dual formulation (4.24) of $\mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}$ . For this, we first note

\begin{equation*} \frac {1}{2}\lVert (u,W) \rVert _{L^2_{\mathsf {G},\Lambda }(Q)}^2 = \sup _{(u^{\prime},W^{\prime}) \in L^2_{\mathsf {G},\Lambda }(Q,\mathbb {R}^{n \times k} \times \mathbb {M}^n)} \langle (u,W), (u^{\prime},W^{\prime})\rangle _{L^2_{\mathsf {G},\Lambda }(Q)} - \frac {1}{2}\lVert (u^{\prime},W^{\prime}) \rVert ^2_{L^2_{\mathsf {G},\Lambda }(Q)} \,, \end{equation*}

which further implies, by $(u_*, W_*) \in H_{\mathsf {G},\Lambda }(\mathsf {D}^*) \subset L^2_{\mathsf {G},\Lambda }(Q,\mathbb {R}^{n \times k} \times \mathbb {M}^n)$ , for any $\Phi \in C^1(Q,\mathbb {S}^n)$ ,

(4.25) \begin{align} \mathcal {J}^{\Lambda }_{\mathsf {G}_0,\mathsf {G}_1}(\mathsf {G}) = \frac {1}{2}\lVert (u_*, W_*) \rVert _{L^2_{\mathsf {G},\Lambda }(Q)}^2 &\ge \langle (u_*, W_*), \big (\mathsf {D}^* \Phi \Lambda ^2_1, \Phi \Lambda _2^2 \big )\rangle _{L^2_{\mathsf {G},\Lambda }(Q)} - \frac {1}{2}\lVert ( \mathsf {D}^* \Phi \Lambda _1^2,\Phi \Lambda _2^2) \rVert ^2_{L^2_{\mathsf {G},\Lambda }(Q)}\notag \\ & = l_G(\Phi ) - \frac {1}{2}\lVert ( \mathsf {D}^* \Phi \Lambda _1^2,\Phi \Lambda _2^2) \rVert ^2_{L^2_{\mathsf {G},\Lambda }(Q)}\,. \end{align}

Then, recalling (4.20) and choosing a sequence $\{(\mathsf {D}^* \Phi _n \Lambda _1^2, \Phi _n \Lambda _2^2)\}$ with $\Phi _n \in C^1(Q,\mathbb {S}^n)$ in (4.25) that approximates $(u_*, W_*)$ gives the desired (4.24).

4.3. Varying weight matrices

We regard $\mathrm {WB}_{\Lambda }$ as a family of distances indexed by $\Lambda$ and investigate the behaviours of $\mathrm {WB}_{\Lambda }$ and its minimiser when $\Lambda$ varies, in particular, when $|\Lambda _1|$ or $|\Lambda _2|$ tends to zero or infinity. We give a partial answer to this question in the following proposition. For ease of exposition, we introduce

(4.26) \begin{align} \mathcal {J}^q_{\Lambda _1}(\mu ) = \mathcal {J}_{\Lambda, Q}(\mathsf {(G,q,0)}) \,, \quad \mathcal {J}^R_{\Lambda _2}(\mu ) = \mathcal {J}_{\Lambda, Q}(\mathsf {(G,0,R)}) \quad \text {for}\ \mu \in \mathcal {M}(Q,\mathbb {X})\,. \end{align}

Proof. We first claim that $\lVert \Lambda _1 \rVert _{\mathrm {F}}^2\mathcal {J}^q_{\Lambda _1}(\mu _{*,\Lambda })$ and $\mathcal {J}^R_{\Lambda _2}(\mu _{*,\Lambda })$ are bounded when $\lVert \Lambda _1 \rVert _{\mathrm {F}} \to 0$ , which, by estimates (3.25) and (3.28), implies that $\mu _{*,\Lambda }$ is bounded in $\mathcal {M}(Q,\mathbb {X})$ . For this, we consider the set

(4.27) \begin{align} \mathcal {CE}_{\Lambda _1, q} \,:\!=\, \arg \min \{\mathcal {J}^q_{\Lambda _1}(\mu )\,; \ \mu \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)\}\,. \end{align}

Similarly to the proof of Lemma3.9, we have that $\mathcal {CE}_{\Lambda _1, q}$ is nonempty and contains at least one element with $\mathsf {q} = 0$ and $\min \{\mathcal {J}^q_{\Lambda _1}(\mu )\,; \ \mu \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)\} = 0$ . Since $\mu _{*,\Lambda }$ minimises $\mathcal {J}_{\Lambda, Q}(\cdot )$ , it follows that

(4.28) \begin{equation} \mathcal {J}_{\Lambda, Q}(\mu _{*,\Lambda }) = \mathcal {J}^q_{\Lambda _1}(\mu _{*,\Lambda }) + \mathcal {J}^R_{\Lambda _2}(\mu _{*,\Lambda })\le \mathcal {J}_{\Lambda, Q}(\mu ) = \mathcal {J}^R_{\Lambda _2}(\mu )\,, \quad \forall \mu = \mathsf {(G,0,R)}\in \mathcal {CE}_{\Lambda _1, q}\,. \end{equation}

Noting $\{\mathsf {(G,0,R)}\in \mathcal {CE}_{\Lambda _1, q}\} = \{\mathsf {(G,0,R)} \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)\}$ , (4.28) yields that $\mathcal {J}^R_{\Lambda _2}(\mu _{*,\Lambda })$ is bounded by a constant independent of $\Lambda _1$ . Moreover, multiplying $\lVert \Lambda _1 \rVert _{\mathrm {F}}^2$ on both sides of (4.28) and then letting $\lVert \Lambda _1 \rVert _{\mathrm {F}} \to 0$ , we obtain

(4.29) \begin{equation} \lim _{\lVert \Lambda _1 \rVert _{\mathrm {F}} \to 0} \lVert \Lambda _1 \rVert _{\mathrm {F}}^2\,\mathcal {J}^q_{\Lambda _1}(\mu _{*,\Lambda }) = 0\,. \end{equation}

Then the boundedness of $\lVert \Lambda _1 \rVert _{\mathrm {F}}^2\mathcal {J}^q_{\Lambda _1}(\mu _{*,\Lambda })$ for small enough $\lVert \Lambda _1 \rVert _{\mathrm {F}}$ follows. We complete the proof of the claim.

By the boundedness of $\lVert \mu _{*,\Lambda } \rVert _{\mathrm {TV}}$ as $\lVert \Lambda _1 \rVert _{\mathrm {F}} \to 0$ , we are allowed to take a subsequence $\{\Lambda _{1,j}\}_{j \in \mathbb {N}}$ in $\mathbb {S}_+^n$ such that the minimiser $\mu _{*,\widetilde {\Lambda }_j}$ with $\widetilde {\Lambda }_j = (\Lambda _{1,j},\Lambda _2)$ weak^* converges to a measure $\mu _* \in \mathcal {M}(Q,\mathbb {X})$ when $n \to \infty$ , which clearly satisfies $\mu _* \in \mathcal {CE}([0,1];\, \mathsf {G}_0,\mathsf {G}_1)$ . Then, by the weak^* lower semicontinuity of $\mathcal {J}^R_{\Lambda _2}$ and (4.28), we have

(4.30) \begin{equation} \mathcal {J}^R_{\Lambda _2}(\mu _*) \le \liminf _{j\to \infty } \mathcal {J}^R_{\Lambda _2}(\mu _{*,\widetilde {\Lambda }_j}) \le \limsup _{j\to \infty } \mathrm {WB}^2_{\widetilde {\Lambda }_j}(\mathsf {G}_0,\mathsf {G}_1) \le \inf \{\mathcal {J}^R_{\Lambda _2}(\mu )\,;\ \mu = \mathsf {(G,0,R)}\in \mathcal {CE}_{\Lambda _1, q}\}\,. \end{equation}

The right-hand side of (4.30) is recognised as $\mathrm {WB}_{(0,\Lambda _2)}(\mathsf {G}_0,\mathsf {G}_1)$ and the inf is attained; see Remark3.11 and Theorem4.2. Also, by (3.25) and (4.29), it holds that the limit measure $\mu _* \in \mathcal {CE}([0,1];\,\mathsf {G}_0,\mathsf {G}_1)$ is of the form $\mu _* = (\mathsf{G}_{*}, \mathsf{0}, \mathsf{R}_{*})$ . The proof is completed by (4.30).

Proposition4.6 above tells us that the measure $\mathsf {q}$ is forced to be nearly zero, if the transportation part is given too much weight (i.e., $\lVert \Lambda _1 \rVert _{\mathrm {F}}$ is small, cf. (3.24)), equivalently, if the problem is on a large scale (cf. Remark3.17). It is also possible and interesting to consider other limiting regimes, e.g., $\lVert \Lambda _1 \rVert _{\mathrm {F}} \to \infty$ , $\lVert \Lambda _2 \rVert _{\mathrm {F}} \to 0$ , or only let part of eigenvalues of $\Lambda _i$ vanish, which, however, is beyond the scope of this work.