SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proposition 10 (Uniform displacement convexity). Let  $\alpha, \beta > 0$ . Fix  $\gamma_0, \gamma_1 \in \mathcal{P}_2 \times \mathcal{P}_2$  and let Assumptions 1-3 hold. Along any geodesic  $(\gamma_s)_{s \in [0,1]} \in \mathcal{P}_2 \times \mathcal{P}_2$  connecting  $\gamma_0$  to  $\gamma_1$ , we have for all  $s \in [0,1]$
$$
\frac {\mathrm {d} ^ {2}}{\mathrm {d} s ^ {2}} G _ {a} \left(\gamma_ {s}\right) \geq \lambda_ {a} \bar {W} \left(\gamma_ {0}, \gamma_ {1}\right) ^ {2}, \quad \lambda_ {a} := \lambda_ {1} + \min  \left(\lambda_ {2} + \beta , a \tilde {\lambda}\right). \tag {15}
$$
As a result, the functional  $G_{a}:\mathcal{P}\times \mathcal{P}\to \mathbb{R}$  is uniformly displacement convex with constant  $\lambda_{a} > 0$ .
Proof. Let  $\gamma_0$  and  $\gamma_{1}$  be two probability measures with bounded second moments. Denote by  $\phi ,\psi :\mathbb{R}^d\to \mathbb{R}$  the optimal Kantorovich potentials pushing  $\rho_0$  onto  $\rho_{1}$ , and  $\mu_0$  onto  $\mu_{1}$ , respectively:
$$
\rho_ {1} = \nabla \phi_ {\#} \rho_ {0} \quad \text {s u c h t h a t} \quad W _ {2} (\rho_ {0}, \rho_ {1}) ^ {2} = \int_ {\mathbb {R} ^ {d}} \| z - \nabla \phi (z) \| ^ {2} \mathrm {d} \rho_ {0} (z),
$$
$$
\mu_ {1} = \nabla \psi_ {\#} \mu_ {0} \quad \text {s u c h t h a t} \quad W _ {2} (\mu_ {0}, \mu_ {1}) ^ {2} = \int_ {\mathbb {R} ^ {d}} \| x - \nabla \psi (x) \| ^ {2} \mathrm {d} \mu_ {0} (x).
$$
The now classical results in [BB00] guarantee that there always exists convex functions  $\phi, \psi$  that satisfy the conditions above. Then the path  $(\gamma_s)_{s \in [0,1]} = (\rho_s, \mu_s)_{s \in [0,1]}$  defined by
$$
\rho_ {s} = \left[ (1 - s) \mathrm {i d} + s \nabla_ {z} \phi \right] _ {\#} \rho_ {0},
$$
$$
\mu_ {s} = \left[ (1 - s) \mathrm {i d} + s \nabla_ {x} \psi \right] _ {\#} \mu_ {0}
$$
is a  $\overline{W}$ -geodesic from  $\gamma_0$  to  $\gamma_1$ .
The first derivative of  $\mathcal{V}$  along geodesics in the Wasserstein metric is given by
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} s} \mathcal {V} (\gamma_ {s}) = \frac {\mathrm {d}}{\mathrm {d} s} \left[ \iint f _ {1} ((1 - s) z + s \nabla \phi (z), (1 - s) x + s \nabla \psi (x)) \mathrm {d} \rho_ {0} (z) \mathrm {d} \mu_ {0} (x) \right. \\ \left. + \int V ((1 - s) x + s \nabla \psi (x)) d \mu_ {0} (x) \right] \\ = \iint \nabla_ {x} f _ {1} ((1 - s) z + s \nabla \phi (z), (1 - s) x + s \nabla \psi (x)) \cdot (\nabla \psi (x) - x) d \rho_ {0} (z) d \mu_ {0} (x) \\ \iint \nabla_ {z} f _ {1} ((1 - s) z + s \nabla \phi (z), (1 - s) x + s \nabla \psi (x)) \cdot (\nabla \phi (z) - z) d \rho_ {0} (z) d \mu_ {0} (x) \\ + \int \nabla_ {x} V ((1 - s) x + s \nabla \psi (x)) \cdot (\nabla \psi (x) - x) d \mu_ {0} (x), \\ \end{array}
$$
and taking another derivative we have
$$
\begin{array}{l} \frac {\mathrm {d} ^ {2}}{\mathrm {d} s ^ {2}} \mathcal {V} (\gamma_ {s}) = - \iint \left[ \begin{array}{c} (\nabla \psi (x) - x) \\ (\nabla \phi (z) - z) \end{array} \right] ^ {T} \cdot D _ {s} (z, x) \cdot \left[ \begin{array}{c} (\nabla \psi (x) - x) \\ (\nabla \phi (z) - z) \end{array} \right]   \mathrm {d} \rho_ {0} (z) \mathrm {d} \mu_ {0} (x) \\ + \iint (\nabla \psi (x) - x) ^ {T} \cdot \nabla_ {x} ^ {2} V ((1 - s) x + s \nabla \psi (x)) \cdot (\nabla \psi (x) - x) \mathrm {d} \rho_ {0} (z) \mathrm {d} \mu_ {0} (x) \\ \geq \lambda_ {1} \bar {W} (\gamma_ {0}, \gamma_ {1}) ^ {2} + (\lambda_ {2} + \beta) W _ {2} (\mu_ {0}, \mu_ {1}) ^ {2}, \\ \end{array}
$$
where we denoted  $D_{s}(z,x)\coloneqq \mathrm{Hess}(f_{1})((1 - s)z + s\nabla \phi (z),(1 - s)x + s\nabla \psi (x))$ , and the last inequality follows from Assumption 1 and the optimality of the potentials  $\phi$  and  $\psi$ .
Following [CMV03; Vil03] and using Assumption 2, the second derivatives of the diffusion term and the interaction term along geodesics are given by
$$
\frac {\mathrm {d} ^ {2}}{\mathrm {d} s ^ {2}} K L \left(\rho_ {s} | \tilde {\rho}\right) \geq \alpha \tilde {\lambda} W _ {2} \left(\rho_ {0}, \rho_ {1}\right) ^ {2}, \quad \frac {\mathrm {d} ^ {2}}{\mathrm {d} s ^ {2}} \mathcal {W} \left(\rho_ {s}\right) \geq 0. \tag {16}
$$
Putting the above estimates together, we obtain (15).
Remark 5. Alternatively, one could assume strong convexity of  $W$ , which would improve the lower-bound on the second derivative along geodesics.
Proposition 11. (Ground state) Let Assumptions 1-3 hold for  $\alpha, \beta > 0$ . Then the functional  $G_{a} : \mathcal{P}(\mathbb{R}^{d}) \times \mathcal{P}(\mathbb{R}^{d}) \to [0, \infty]$  admits a unique minimizer  $\gamma_{*} = (\rho_{*}, \mu_{*})$ , and it satisfies  $\rho_{*} \in \mathcal{P}_{2}(\mathbb{R}^{d}) \cap L^{1}(\mathbb{R}^{d})$ ,  $\mu_{*} \in \mathcal{P}_{2}(\mathbb{R}^{d})$ , and  $\rho_{*}$  is absolutely continuous with respect to  $\tilde{\rho}$ .
Proof. We show existence of a minimizer of  $G_{a}$  using the direct method in the calculus of variations. Denote by  $\gamma = (\rho, \mu) \in \mathcal{P} \times \mathcal{P} \subset \mathcal{M} \times \mathcal{M}$  a pair of probability measures as a point in the product space of Radon measures. Since  $G_{a} \geq 0$  on  $\mathcal{P} \times \mathcal{P}$  (see Assumption 1) and not identically  $+\infty$  everywhere, there exists a minimizing sequence  $(\gamma_{n}) \in \mathcal{P} \times \mathcal{P}$ . Note that  $(\gamma_{n})$  is in the closed unit ball of the dual space of continuous functions vanishing at infinity  $(C_0(\mathbb{R}^d) \times C_0(\mathbb{R}^d))^*$  endowed with the dual norm  $\| \gamma_{n} \|_{*} = \sup \frac{\left| \int f \, \mathrm{d}\rho_{n} + \int g \, \mathrm{d}\mu_{n} \right|}{\left\| (f, g) \right\|_{\infty}}$  over  $f, g \in C_0(\mathbb{R}^d)$  with  $\| (f, g) \|_{\infty} := \| f \|_{\infty} + \| g \|_{\infty} \neq 0$ . By the Banach-Alaoglu theorem [Rud91, Thm 3.15] there exists a limit  $\gamma_{*} = (\rho_{*}, \mu_{*}) \in \mathcal{M} \times \mathcal{M} = (C_0 \times C_0)^*$  and a convergent subsequence (not relabelled) such that  $\gamma_{n} \stackrel{*}{\to} \gamma_{*}$ . In fact, since  $KL(\rho_{*} | \tilde{\rho}) < \infty$  it follows that  $\rho_{*}$  is absolutely continuous with respect to  $\tilde{\rho}$ , implying  $\rho_{*} \in L^{1}(\mathbb{R}^{d})$  thanks to Assumption 2. Further,  $\mu_{*}$  has bounded second moment, else we would have
$\inf_{\gamma \in \mathcal{P} \times \mathcal{P}} G_{a}(\gamma) = \infty$  which yields a contradiction. It remains to show that  $\int \mathrm{d}\rho_{*} = \int \mathrm{d}\mu_{*} = 1$  to conclude that  $\gamma_{*} \in \mathcal{P} \times \mathcal{P}$ . To this aim, it is sufficient to show tightness of  $(\rho_{n})$  and  $(\mu_{n})$ , preventing the escape of mass to infinity as we have  $\int \mathrm{d}\rho_{n} = \int \mathrm{d}\mu_{n} = 1$  for all  $n \geq 1$ . Tightness follows from Markov's inequality [Gho02] if we can establish uniform bounds on the second moments, i.e. we want to show that there exists a constant  $C > 0$  independent of  $n$  such that
$$
\int \| z \| ^ {2} \mathrm {d} \rho_ {n} (z) + \int \| x \| ^ {2} \mathrm {d} \mu_ {n} (x) <   C \quad \forall n \in \mathrm {N}. \tag {17}
$$
To establish (17), observe that thanks to Assumption 2, there exists a constant  $c_{0} \in \mathbb{R}$  (possibly negative) such that  $-\log \tilde{\rho}(z) \geq c_{0} + \frac{\tilde{\lambda}}{4} \|z\|^{2}$  for all  $z \in \mathbb{R}^{d}$ . Then
$$
\frac {\alpha \tilde {\lambda}}{4} \int \| z \| ^ {2} \mathrm {d} \rho_ {n} \leq - \alpha c _ {0} - \alpha \int \log \tilde {\rho} (z) \mathrm {d} \rho_ {n}
$$
Therefore, using  $\int \mathrm{d}\rho_{n} = \int \mathrm{d}\mu_{n} = 1$  and writing  $\zeta \coloneqq \min \left\{\frac{\alpha\lambda}{4},\frac{\beta}{2}\right\} >0$ , we obtain the desired uniform upper bound on the second moments of the minimizing sequence,
$$
\begin{array}{l} \zeta \iint \left(\| z \| ^ {2} + \| x \| ^ {2}\right) \mathrm {d} \rho_ {n} \mathrm {d} \mu_ {n} \leq - \alpha c _ {0} - \alpha \int \log \tilde {\rho} (z) \mathrm {d} \rho_ {n} + \beta \int \| x - x _ {0} \| ^ {2} \mathrm {d} \mu_ {n} + \beta \| x _ {0} \| ^ {2} \\ \leq - \alpha c _ {0} + \beta \| x _ {0} \| ^ {2} + G _ {a} (\gamma_ {n}) \\ \leq - \alpha c _ {0} + \beta \| x _ {0} \| ^ {2} + G _ {a} (\gamma_ {1}) <   \infty . \\ \end{array}
$$
This concludes the proof that the limit  $\gamma_{*}$  satisfies indeed  $\gamma_{*} \in \mathcal{P} \times \mathcal{P}$ , and indeed  $\rho_{*} \in \mathcal{P}_{2}(\mathbb{R}^{d})$  as well. Finally,  $\gamma_{*}$  is indeed a minimizer of  $G_{a}$  thanks to weak- $^{*}$  lower-semicontinuity of  $G_{a}$  following Lemma 9.
Next we show uniqueness using a contradiction argument. Suppose  $\gamma_{*} = (\rho_{*},\mu_{*})$  and  $\gamma_{*}^{\prime} = (\rho_{*}^{\prime},\mu_{*}^{\prime})$  are minimizers of  $G_{a}$ . For  $s\in [0,1]$ , define  $\gamma_s\coloneqq ((1 - s)\mathrm{id} + sT,(1 - s)\mathrm{id} + sS)_{\#}\gamma_{*}$ , where  $T,S:\mathbb{R}^{d}\mapsto \mathbb{R}^{d}$  are the optimal transport maps such that  $\rho_{*}^{\prime} = T_{\#}\rho_{*}$  and  $\mu_{*}^{\prime} = S_{\#}\mu_{*}$ . By Proposition 10 the energy  $G_{a}$  is uniformly displacement convex, and so we have
$$
G _ {a} (\gamma_ {s}) \leq (1 - s) G _ {a} (\gamma_ {*}) + s G _ {a} \left(\gamma_ {*} ^ {\prime}\right) = G _ {a} (\gamma_ {*}).
$$
If  $\gamma_{*} \neq \gamma_{*}^{\prime}$  and  $s \in (0,1)$ , then strict inequality holds by applying similar arguments as in [McC97, Proposition 1.2]. However, the strict inequality  $G_{a}(\gamma_{s}) < G_{a}(\gamma_{*})$  for  $\gamma_{*} \neq \gamma_{*}^{\prime}$  is a contradiction to the minimality of  $\gamma_{*}$ . Hence, the minimizer is unique.
Remark 6. If  $\lambda_1 > 0$ , then the strict convexity of  $f_{1}$  can be used to deduce uniqueness, and the assumptions on  $-\log \tilde{\rho}$  can be weakened from strict convexity to convexity.
Corollary 12. Let Assumptions 1-3 hold. Any minimizer  $\gamma_{*} = (\rho_{*},\mu_{*})$  of  $G_{a}$  is a steady state for equation (4) according to Definition 8 and satisfies  $\mathrm{supp}(\rho_{*}) = \mathrm{supp}(\tilde{\rho})$ .
Proof. By Proposition 11, we have  $\rho_{*},\mu_{*}\in \mathcal{P}_{2}$ , as well as  $\rho_{*}\in L_{+}^{1}$ $\| \rho_{*}\|_{1} = 1$  and that  $\rho_{*}$  is absolutely continuous with respect to  $\tilde{\rho}$ . Since  $W\in C^2 (\mathbb{R}^d)$ , it follows that  $\nabla W*\rho_{*}\in L_{loc}^{1}$ . In order to show that  $\gamma_{*}$  is a steady state for equation (4), it remains to prove that  $\rho_{*}\in W_{loc}^{1,2}\cap L_{loc}^{\infty}$ . As  $\gamma^{*}$  is a minimizer, it is in particular a critical point, and therefore satisfies equations (14). Rearranging, we obtain (for a possible different constant  $c_{1}[\rho_{*},\mu_{*}]\neq 0$ ) from (14a) that
$$
\rho_ {*} (z) = c _ {1} \left[ \rho_ {*}, \mu_ {*} \right] \tilde {\rho} (z) \exp \left[ - \frac {1}{\alpha} \left(\int f _ {1} (z, x) \mu_ {*} (x) + W * \rho_ {*} (z)\right) \right] \quad \text {o n}
$$
Then for any compact set  $K\subset \mathbb{R}^d$
$$
\sup  _ {z \in K} \rho_ {*} (z) \leq c _ {1} [ \rho_ {*}, \mu_ {*} ] \sup  _ {z \in K} \tilde {\rho} (z) \sup  _ {z \in K} \exp \left(- \frac {1}{\alpha} \left(\int f _ {1} (z, x) \mu_ {*} (x)\right)\right) \sup  _ {z \in K} \exp \left(- \frac {1}{\alpha} W * \rho_ {*}\right).
$$
As  $f_{1} \geq 0$  by Assumption 1 and  $W \geq 0$  by Assumption 3, the last two terms are finite. The first supremum is finite thanks to continuity of  $\tilde{\rho}$ . Therefore  $\rho_{*} \in L_{loc}^{\infty}$ . To show that  $\rho_{*} \in W_{loc}^{1,2}$ , note that for any compact set  $K \subset \mathbb{R}^d$ , we have  $\int_K |\rho_*(z)|^2 \mathrm{d}z < \infty$  as a consequence of  $\rho_{*} \in L_{loc}^{\infty}$ . Moreover, defining  $T[\gamma](z) := -\frac{1}{\alpha} \left( \int f_1(z,x) \mu(x) + W * \rho(z) \right) \leq 0$ , we have
$$
\begin{array}{l} \int_ {K} | \nabla \rho_ {*} | ^ {2} d z = c _ {1} [ \rho_ {*}, \mu_ {*} ] ^ {2} \int_ {K} | \nabla \tilde {\rho} + \tilde {\rho} \nabla T [ \gamma_ {*} ] | ^ {2} \exp (2 T [ \gamma_ {*} ]) d z \\ \leq 2 c _ {1} [ \rho_ {*}, \mu_ {*} ] ^ {2} \int_ {K} | \nabla \tilde {\rho} | ^ {2} \exp (2 T [ \gamma_ {*} ]) \mathrm {d} z + 2 c _ {1} [ \rho_ {*}, \mu_ {*} ] ^ {2} \int_ {K} | \nabla T [ \gamma_ {*} ] | ^ {2} \tilde {\rho} ^ {2} \exp (2 T [ \gamma_ {*} ]) \mathrm {d} z, \\ \end{array}
$$
which is bounded noting that  $\exp (2T[\gamma_{*}])\leq 1$  and that  $T[\gamma_{*}](\cdot),\nabla T[\gamma_{*}](\cdot)$  and  $\nabla \tilde{\rho}$  are in  $L_{loc}^{\infty}$ , where we used that  $f_{1},(\cdot ,x),W(\cdot),\tilde{\rho} (\cdot)\in C^{1}(\mathbb{R}^{d})$  by Assumptions 1-3. We conclude that  $\rho_{*}\in W_{loc}^{1,2}$ , and indeed  $(\rho_{*},\mu_{*})$  solves (12) in the sense of distributions as a consequence of (14).
Next, we show that  $\mathrm{supp}(\rho_{*}) = \mathrm{supp}(\tilde{\rho})$  using again the relation (18). Firstly, note that  $\mathrm{supp}(\rho_{*})\subset \mathrm{supp}(\tilde{\rho})$  since  $\rho_{*}$  is absolutely continuous with respect to  $\tilde{\rho}$ . Secondly, we claim that  $\exp \left[-\frac{1}{\alpha}\left(\int f_1(z,x)\mu_*(x) + W*\rho_*(z)\right)\right] > 0$  for all  $z\in \mathbb{R}^d$ . In other words, we claim that  $\int f_1(z,x)\mu_*(x) < \infty$  and  $W*\rho_{*}(z) < \infty$  for all  $z\in \mathbb{R}^d$ . Indeed, for the first term, fix any  $z\in \mathbb{R}^d$  and choose  $R > 0$  large enough such that  $z\in B_R(0)$ . Then, thanks to continuity of  $f_{1}$  according to Assumption 1, we have
$$
\int f _ {1} (z, x) \mu_ {*} (x) \leq \sup  _ {z \in B _ {R} (0)} \int f _ {1} (z, x) \mu_ {*} (x) <   \infty .
$$
For the second term, note that by Assumption 3, we have for any  $z \in \mathbb{R}^d$  and  $\epsilon > 0$ ,
$$
\begin{array}{l} W (z) \leq W (0) + \nabla W (z) \cdot z \leq W (0) + \frac {1}{2 \epsilon} \| \nabla W (z) \| ^ {2} + \frac {\epsilon}{2} \| z \| ^ {2} \\ \leq W (0) + \frac {D ^ {2}}{2 \epsilon} \left(1 + \| z \|\right) ^ {2} + \frac {\epsilon}{2} \| z \| ^ {2} \leq W (0) + \frac {D ^ {2}}{\epsilon} + \left(\frac {D ^ {2}}{\epsilon} + \frac {\epsilon}{2}\right) \| z \| ^ {2} \\ = W (0) + \frac {D}{\sqrt {2}} + \sqrt {2} D \| z \| ^ {2}, \\ \end{array}
$$
where the last equality follows by choosing the optimal  $\epsilon = \sqrt{2} D$ . We conclude that
$$
\begin{array}{l} W * \rho_ {*} (z) \leq W (0) + \frac {D}{\sqrt {2}} + \sqrt {2} D \int \| z - \tilde {z} \| ^ {2} \rho_ {*} (\tilde {z}) \\ \leq W (0) + \frac {D}{\sqrt {2}} + 2 \sqrt {2} D \| z \| ^ {2} + 2 \sqrt {2} D \int \| \tilde {z} \| ^ {2} \rho_ {*} (\tilde {z}), \tag {19} \\ \end{array}
$$
which is finite for any fixed  $z \in \mathbb{R}^d$  thanks to the fact that  $\rho_* \in \mathcal{P}_2(\mathbb{R}^d)$ . Hence,  $\operatorname{supp}(\rho_*) = \operatorname{supp}(\tilde{\rho})$ .
Remark 7. If we have in addition that  $\tilde{\rho} \in L^{\infty}(\mathbb{R}^{d})$ , then the minimizer  $\rho_{*}$  of  $G_{a}$  is in  $L^{\infty}(\mathbb{R}^{d})$  as well. This follows directly by bounding the right-hand side of (18).
The following inequality is referred to as HWI inequality and represents the key result to obtain convergence to equilibrium.
Proposition 13 (HWI inequality). Define the dissipation functional
$$
D _ {a} (\gamma) := \iint | \nabla_ {x, z} \delta_ {\gamma} G _ {a} (z, x) | ^ {2} \mathrm {d} \gamma (z, x).
$$
Assume  $\alpha, \beta > 0$  and let  $\lambda_{a}$  as defined in (15). Let  $\gamma_0, \gamma_1 \in \mathcal{P}_2 \times \mathcal{P}_2$  such that  $G_{a}(\gamma_{0}), G_{a}(\gamma_{1}), D_{a}(\gamma_{0}) < \infty$ . Then
$$
G _ {a} (\gamma_ {0}) - G _ {a} (\gamma_ {1}) \leq \bar {W} (\gamma_ {0}, \gamma_ {1}) \sqrt {D _ {a} (\gamma_ {0})} - \frac {\lambda_ {a}}{2} \bar {W} (\gamma_ {0}, \gamma_ {1}) ^ {2} \tag {20}
$$
Proof. For simplicity, consider  $\gamma_0,\gamma_1$  that have smooth Lebesgue densities of compact support. The general case can be recovered using approximation arguments. Let  $(\gamma_s)_{s\in [0,1]}$  denote a  $\overline{W}$ -geodesic between  $\gamma_0,\gamma_1$ . Following similar arguments as in [CMV03] and [OV00, Section 5] and making use of the calculations in the proof of Proposition 10, we have
$$
\left. \frac {\mathrm {d}}{\mathrm {d} s} G _ {a} (\gamma_ {s}) \right| _ {s = 0} \geq \iint \left[ \begin{array}{c} \xi_ {1} (z) \\ \xi_ {2} (x) \end{array} \right] \cdot \left[ \begin{array}{c} (\nabla \phi (z) - z) \\ (\nabla \psi (x) - x) \end{array} \right]   \mathrm {d} \gamma_ {0} (z, x),
$$
$$
\begin{array}{l} \xi_ {1} \left[ \gamma_ {0} \right] (z) := \alpha \nabla_ {z} \log \left(\frac {\rho_ {0} (z)}{\tilde {\rho} (z)}\right) + \int \nabla_ {z} f _ {1} (z, x) d \mu_ {0} (x) + \int \nabla_ {z} W (z - z ^ {\prime}) d \rho_ {0} \left(z ^ {\prime}\right), \\ \xi_ {2} [ \gamma_ {0} ] (x) := \int \nabla_ {x} f _ {1} (z, x) d \rho_ {0} (z) + \nabla_ {x} V (x). \\ \end{array}
$$
Note that the dissipation functional can then be written as
$$
D _ {a} (\gamma_ {0}) = \iint \left(| \xi_ {1} (z) | ^ {2} + | \xi_ {2} (x) | ^ {2}\right) d \gamma_ {0} (z, x).
$$
Using the double integral Cauchy-Schwarz inequality [Ste04], we obtain
$$
\begin{array}{l} \left. \frac {\mathrm {d}}{\mathrm {d} s} G _ {a} (\gamma_ {s}) \right| _ {s = 0} \geq - \left(\sqrt {\iint \left\| \left[ \begin{array}{c} \xi_ {1} \\ \xi_ {2} \end{array} \right] \right\| _ {2} ^ {2} \mathrm {d} \gamma_ {0}}\right) \left(\sqrt {\iint \left\| \left[ \begin{array}{c} \nabla \phi (z) - z \\ \nabla \psi (x) - x \end{array} \right] \right\| _ {2} ^ {2} \mathrm {d} \gamma_ {0}}\right) \\ = - \sqrt {D _ {a} (\gamma_ {0})} \sqrt {\int \| \nabla \phi (z) - z \| ^ {2} \mathrm {d} \rho_ {0} + \int \| \nabla \psi (x) - x \| ^ {2} \mathrm {d} \mu_ {0}} \\ = - \sqrt {D _ {a} (\gamma_ {0})} \bar {W} (\gamma_ {0}, \gamma_ {1}). \\ \end{array}
$$
Next, we compute a Taylor expansion of  $G_{a}(\gamma_{s})$  when considered as a function in  $s$  and use the bound on  $\frac{\mathrm{d}^2}{\mathrm{d}s^2} G_{a}$  from (15):
$$
\begin{array}{l} G _ {a} (\gamma_ {1}) = G _ {a} (\gamma_ {0}) + \left. \frac {\mathrm {d}}{\mathrm {d} s} G _ {a} (\gamma_ {s}) \right| _ {s = 0} + \int_ {0} ^ {1} (1 - t) \left(\frac {\mathrm {d} ^ {2}}{\mathrm {d} s ^ {2}} G _ {a} (\gamma_ {s})\right) \Bigg | _ {s = t} \mathrm {d} t \\ \geq G _ {a} \left(\gamma_ {0}\right) - \sqrt {D _ {a} \left(\gamma_ {0}\right)} \bar {W} \left(\gamma_ {0}, \gamma_ {1}\right) + \frac {\lambda_ {a}}{2} \bar {W} \left(\gamma_ {0}, \gamma_ {1}\right) ^ {2}. \\ \end{array}
$$
Remark 8. The HWI inequality in Proposition 13 immediately implies uniqueness of minimizers for  $G_{a}$  in the set  $\{\gamma \in \mathcal{P} \times \mathcal{P} : D_{a}(\gamma) < +\infty\}$ . Indeed, if  $\gamma_{0}$  is such that  $D_{a}(\gamma_{0}) = 0$ , then for any other  $\gamma_{1}$  in the above set we have  $G_{a}(\gamma_{0}) \leq G_{a}(\gamma_{1})$  with equality if and only if  $\overline{W}(\gamma_{0}, \gamma_{1}) = 0$ .
Corollary 14 (Generalized Log-Sobolev inequality). Denote by  $\gamma_*$  the unique minimizer of  $G_a$ . With  $\lambda_a$  as defined in (15), any product measure  $\gamma \in \mathcal{P}_2 \times \mathcal{P}_2$  such that  $G(\gamma)$ ,  $D_a(\gamma) < \infty$  satisfies
$$
D _ {a} (\gamma) \geq 2 \lambda_ {a} G _ {a} (\gamma | \gamma_ {*}). \tag {21}
$$
Proof. This statement follows immediately from Proposition 13. Indeed, let  $\gamma_{1} = \gamma_{*}$  and  $\gamma_0 = \gamma$  in (20). Then
$$
\begin{array}{l} G _ {a} (\gamma \mid \gamma_ {*}) \leq \overline {{W}} (\gamma , \gamma_ {*}) \sqrt {D _ {a} (\gamma)} - \frac {\lambda_ {a}}{2} \overline {{W}} (\gamma , \gamma_ {*}) ^ {2} \\ \leq \max _ {t \geq 0} \left(\sqrt {D _ {a} (\gamma)} t - \frac {\lambda_ {a}}{2} t ^ {2}\right) = \frac {D _ {a} (\gamma)}{2 \lambda_ {a}}. \\ \end{array}
$$
Corollary 15 (Talagrand inequality). Denote by  $\gamma_{*}$  the unique minimizer of  $G_{a}$ . With  $\lambda_{a}$  as defined in (15), it holds
$$
\overline {{W}} (\gamma , \gamma_ {*}) ^ {2} \leq \frac {2}{\lambda_ {a}} G _ {a} (\gamma | \gamma_ {*})
$$
for any  $\gamma \in \mathcal{P}_2\times \mathcal{P}_2$  such that  $G_{a}(\gamma) <   \infty$
Proof. This is also a direct consequence of Proposition 13 by setting  $\gamma_0 = \gamma_*$  and  $\gamma_1 = \gamma$ . Then  $G_{a}(\gamma_{*}) < \infty$  and  $D_{a}(\gamma_{*}) = 0$ , and the result follows.
Proof of Theorem 2. The entropy term  $\int \rho \log \rho$  produces diffusion in  $\rho$  for the corresponding PDE in (4). As a consequence, solutions  $\rho_{t}$  to (4) and minimizers  $\rho^{*}$  for  $G_{a}$  have to be  $L^1$  functions. As there is no diffusion for the evolution of  $\mu_t$ , solutions may have a singular part. In fact, for initial condition  $\mu_0 = \delta_{x_0}$ , the corresponding solution will be of the form  $\mu_t = \delta_{x(t)}$ , where  $x(t)$  solves the ODE (3) with initial condition  $x_0$ . This follows from the fact that the evolution for  $\mu_t$  is a transport equation (also see Section A.1 for more details). Results (a) and (b) are the statements in Proposition 11, Corollary 12 and Corollary 15. To obtain (c), we differentiate the energy  $G_{a}$  along solutions  $\gamma_t$  to the equation (4):
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} G _ {a} (\gamma_ {t}) = \int \delta_ {\rho} G _ {a} [ \gamma_ {t} ] (z) \partial_ {t} \rho_ {t} \mathrm {d} z + \int \delta_ {\mu} G _ {a} [ \gamma_ {t} ] (x) \partial_ {t} \mu_ {t} \mathrm {d} x \\ = - \int \| \nabla_ {z} \delta_ {\rho} G _ {a} [ \gamma_ {t} ] (z) \| ^ {2} d \rho_ {t} (z) - \int \| \nabla_ {x} \delta_ {\mu} G _ {a} [ \gamma_ {t} ] (x) \| ^ {2} d \mu_ {t} (x) \\ = - D _ {a} \left(\gamma_ {t}\right) \leq - 2 \lambda_ {a} G _ {a} \left(\gamma_ {t} \mid \gamma_ {*}\right), \\ \end{array}
$$
where the last bound follows from Corollary 14. Applying Gronwall's inequality, we immediately obtain decay in energy,
$$
G _ {a} \left(\gamma_ {t} \mid \gamma_ {*}\right) \leq e ^ {- 2 \lambda_ {a} t} G _ {a} \left(\gamma_ {0} \mid \gamma_ {*}\right).
$$
Finally, applying Talagrand's inequality (Corollary 15), the decay in energy implies decay in the product Wasserstein metric,
$$
\overline {{W}} (\gamma_ {t}, \gamma_ {*}) \leq c e ^ {- \lambda_ {a} t}
$$
where  $c > 0$  is a constant only depending on  $\gamma_0$ ,  $\gamma_*$  and the parameter  $\lambda_{a}$ .
In the case of competing objectives, we rewrite the energy  $G_{c}(\rho ,x):\mathcal{P}(\mathbb{R}^{d})\times \mathbb{R}^{d}\mapsto [-\infty ,\infty ]$  as follows:
$$
G _ {c} (\rho , x) = \int f _ {1} (z, x) \mathrm {d} \rho (z) + \int f _ {2} (z, x) \mathrm {d} \bar {\rho} (z) + \frac {\beta}{2} \| x - x _ {0} \| ^ {2} - P (\rho),
$$
$$
P (\rho) := \alpha K L (\rho | \tilde {\rho}) + \frac {1}{2} \int \rho W * \rho .
$$
Note that for any fixed  $\rho \in \mathcal{P}$ , the energy  $G_{c}(\rho, \cdot)$  is strictly convex in  $x$ , and therefore has a unique minimizer. Define the best response by
$$
b (\rho) := \underset {\bar {x}} {\operatorname {a r g m i n}} G _ {c} (\rho , \bar {x})
$$
and denote  $G_{b}(\rho) \coloneqq G_{c}(\rho, b(\rho))$ . We begin with auxiliary results computing the first variations of the best response  $b$  and then the different terms in  $G_{b}(\rho)$  using Definition 1.
Lemma 16 (First variation of the best response). The first variation of the best response of the classifier at  $\rho$  (if it exists) is
$$
\delta_ {\rho} b [ \rho ] (z) = - Q (\rho) ^ {- 1} \nabla_ {x} f _ {1} (z, b (\rho)) \quad f o r a l m o s t e v e r y z \in \mathbb {R} ^ {d},
$$
where  $Q(\rho) \succeq (\beta + \lambda_1 + \lambda_2) \mathrm{I}_d$  is a symmetric matrix, constant in  $z$  and  $x$ , defined as
$$
Q (\rho) := \beta I _ {d} + \int \nabla_ {x} ^ {2} f _ {1} (z, b (\rho)) \mathrm {d} \rho (z) + \int \nabla_ {x} ^ {2} f _ {2} (z, b (\rho)) \mathrm {d} \bar {\rho} (z).
$$
In particular, we then have for any  $\psi \in C_c^\infty (\mathbb{R}^d)$  with  $\int \psi \mathrm{d}z = 0$  that
$$
\lim  _ {\epsilon \rightarrow 0} \frac {1}{\epsilon} \left\| b [ \rho + \epsilon \psi ] - b [ \rho ] - \epsilon \int \delta_ {\rho} b [ \rho ] (z) \psi (z) \mathrm {d} z \right\| = 0.
$$
Proof. Let  $\psi \in C_c^\infty (\mathbb{R}^d)$  with  $\int \psi \mathrm{d}z = 0$  and fix  $\epsilon >0$ . Any minimizer of  $G_{c}(\rho +\epsilon \psi ,x)$  for fixed  $\rho$  must satisfy